LAS VEGAS – Even more astounding than it would be to see a hockey coach lift the Stanley Cup and fail to ink a contract extension — a very real, very bizarre possibility — is the manner in which that coach is approaching what should be the most agonizing games of his life.

Only four National Hockey League head coaches in history have more regular-season wins than the 762 belonging to Barry Trotz of the Washington Capitals: Scotty Bowman, Joel Quenneville, Ken Hitchcock and Al Arbour – two Hall of Famers and two that will eventually be.

The difference? Trotz doesn’t have a Stanley Cup ring, yet.

Heck, his 19 clubs hadn’t survived beyond the second round until this one, the one that can guzzle champagne and dive headfirst into a self-inflicted Vegas flu as early as Thursday night if they can win Game 5.

On the surface, it looks crazy that the leader of the Capitals’ best self is also a lame duck.

So it’s even more impressive, then, that the 55-year-old Trotz can be seen playfully cross-checking players into the boards at practice, mimicking Gatorade ads during his press conferences, cracking jokes, and ripping through the club’s superstitious hot lap.

“I went through a few things last summer that gave me some real good clarity on how I define myself or how I define people. It’s given me a real clarity. This is a time to enjoy,” Trotz said Wednesday. “Live in the moment.”

This is Zen Trotz, the most relaxed Nicklas Backstrom says he’s ever seen his boss. Some of Trotz’s best coaching friends have never gone this far. He knows he may never be here again.

“So why make it tense and stressful and all that? You have to respect the process and respect what you need to do to be successful, but enjoy it,” Trotz went on, smiling all the while. “This is not going to happen too often. It’s taken a long time to get here.”

Trotz has kept the details of what he personally endured last summer private. What we do know is this:

In the summer of 2016, Capitals assistants Todd Reirden and Lane Lambert were finalists for the head jobs in Calgary and Avalanche, respectively.

Reirden was then promoted from assistant to associate coach and given increased responsibilities beyond managing the defence. He signed an extension that lasts beyond Trotz’s deal. He had more of an influence on the Capitals’ lethal power-play scheme and ran training camp while Trotz served on Mike Babcock’s gold-winning Team Canada bench at the World Cup.

“You never like to lose high-quality people and coaches, but at the same time these are guys that if they’re not replacing me, they’re replacing someone else in the league,” Trotz told reporters at that time. “That’s what we want. We want to develop people.”

Matt Niskanen, whose trust with the D coach goes back to their Pittsburgh days, went on record saying that Reirden is “probably going to get a chance” to be an NHL bench boss.

“Barry does a great job of empowering us and giving us opportunities to have our own areas of specialization and teach and coach that way. I think that’s why we’re able to garner the attention that we did receive out there from other teams.” Reirden told The Washington Post in 2016.

In the summer of 2017, the Presidents’ Trophy–winning Capitals were ousted in Round 2 by playoff nemesis Pittsburgh for the second consecutive spring. There were rumblings that Trotz’s decision to dress seven defencemen when things started going sideways in the Penguins series did not sit well with everybody. Reirden was reportedly prevented from interviewing for the head jobs in Florida and Buffalo.

Trotz, who was hired prior to GM Brian MacLellan’s appointment, was not awarded an extension entering 2017-18.

When the subject was raised heading into this post-season, Trotz said, “I haven’t lost sleep about it. That’s a Mac question or an ownership question. I could care less about that question other than that’s a question for them.”

MacLellan refused to say whether playoff success would determine Trotz’s fate and that a decision would be made when it’s all over.

“He’s done a good job,” MacLellan said in April. “We were fragile coming in, and it’s not an easy thing to stabilize that.

“To get [the players] to the point where they have the energy and the mental strength to compete again is a big deal. I think he’s been a big part of it, the assistant coaches have been a big part of it, and the players themselves have done a good job.”

Owner Ted Leonsis declined comment to The Post.

On a coaching level, the situation feels similar to what just went down with Kyle Dubas, Mark Hunter and Toronto Maple Leafs management: there’s no way to keep everyone happy long-term.

And yet, winning solves all. It can change perceptions if not minds.

“There was definitely a lot of tension [last spring],” star defenceman John Carlson says. “Breaking through like we did against Pittsburgh was not our goal of the season, but it gave us a good, laid-back feeling in terms of that was the pressure on us, not the rest of us. [Trotz] did a lot better job of that, and it’s trickled down to the players and put us in a pretty good mental space to go out and execute.”

Ever since Trotz departed his beloved Nashville as the longest-tenured coach in the business and took the reins from Adam Oates in 2014, veteran Jay Beagle says an “unreal” difference occurred. Trotz instilled a buy-in to a true team approach and pushed the players’ work ethic.

“He’s brought a culture change. [This is] the most thorough coaching staff I’ve ever been a part of. We’re prepared for everything. It makes the game easier. It makes it so we can go out and do our job,” Beagle explains.

“It’s almost like we know what teams are going to do before they do it, just because we go over it so much and the coaching staff is so thorough.”

During a radio interview Wednesday, former Capitals defenceman Karl Alzner said Trotz is the most inclusive and family-oriented coach he’s ever skated for. Trainers, wives, coaches, they’re all members of the Caps crew. He tries to give everyone Sundays off, holds a keen pulse on the room through his leadership core, and is wise enough to know that if the team is happy away from the rink, they’ll play better on it.

Plus, Alzner said, Washington has “three of the best assistant coaches I’ve ever had. They’ve got a good setup there, and Barry’s the backbone.”

Above everything, Tom Wilson argues, Trotz is a decent man who makes an effort to establish a trust with his players off the ice first.

“The rest is a bonus,” Wilson says. “He came here four years ago and said he wanted a family atmosphere, he wanted everyone to be close. That’s what he preached to us: a team identity. It’s something this team really hadn’t had.”

Beagle, a pending UFA himself, doesn’t even want to consider the idea that Thursday might be Trotz’s last behind this magical team’s bench.

“I might not be here. There’s a lot of guys who might not be here. You never know. You can’t control, so you never think about it,” Beagle says. “Live in the moment.”

Not since the infamous Mike Keenan controversy in New York in 1994 has the Cup-winning coach took his ring and bounced out of town.

There’s a job opening with the Islanders. Maybe Reirden gets it. Maybe Trotz does. Regardless, a Cup ring will sparkle on both their resumes, and hockey families — tight as they grow — are temporary.

“Maybe that’s why he’s so relaxed,” wonders Backstrom. “He doesn’t know what’s going to happen. I don’t think anyone knows.”