It’s a barking shame.

A heroic Belgian Malinois who chased down Abu Bakr al-Baghdadi last week and was wounded after the ISIS boss blew himself up with a suicide vest will not receive a Purple Heart.

While Conan received high praise from President Trump and is expected to visit the White House next week, “military working dogs are not eligible for the Purple Heart,” confirmed Department of Defense spokeswoman Jessica Maxwell.

Multiple human Purple Heart recipients and combat vets who’ve worked closely with K9 troops told The Post that they were outraged by the snub — saying courageous canines like Conan were the reason many of them are alive.

“Do I believe Conan should receive a Purple Heart for actions on target? Absolutely,” said Michael Bollinger, a former Army Ranger now studying computer science at Columbia University. “They’re out there with us every step of the way.”

Bollinger, 26, received a Purple Heart in 2017 after ISIS fighters used a drone to drop a grenade on him and several other Rangers during a deployment to a classified location. He took heavy shrapnel wounds and significant injuries to his legs.

“I am still trying to recover from that,” he said. “Learning to run again has been a challenge.”

Service dogs have been used by the U.S. military for decades. On the battlefields of Iraq and Afghanistan, the animals’ most critical function has been sniffing out hidden roadside bombs or IEDs — and they have saved untold numbers of American lives.

“One explosive can take out six to eight guys,” said Kit Sawyer, a 30-year-old who served as a sergeant and dog handler in the Army’s 75th Ranger regiment, recalling multiple moments when his Dutch Shepard Roy sounded the alarm about the hidden killers in Afghanistan. “He’s definitely saved lives.”

The issue of military dogs receiving Purple Hearts has roiled the armed forces for more than a half-century, when debate first rose in 1943 over World War II hero Chips, a Shepherd-Collie-Husky, who was injured taking out a Nazi gun nest in Sicily. The action was considered so valorous that Chips received not just a Purple Heart but a Silver Star and the Distinguished Service Cross, according to accounts from the time.

Publicity surrounding the matter eventually made its way to William Thomas, then serving as national commander of the Military Order of the Purple Heart. He thought the medals were an insult to their human recipients and said as much in angry letters to President Franklin Roosevelt. In 1944 Major General James Ulio ruled that Chips could keep the medals but no further awards would be handed out to four-legged soldiers. (The edict doesn’t appear to have been enforced too strictly at first; during the Korean War a horse named Sgt. Reckless was given two Purple Hearts.)

But many modern Purple Heart vets believe such views toward combat dogs are outdated.

“I think that a lot of things have changed. Animals have come into humans lives a lot differently than they did 20 years ago,” said James Knuppenburg, 33, who spent nearly 10 years in Afghanistan. “They have become a companion, not just for the battlefield, but also to reverse PTSD and to help men and women with battlefield injuries that are not visible.”

Knuppenburg received his Purple Heart after being shot by a sniper in Kandahar City midway through his five tours. His medic died trying to rescue him and he spent years in physical therapy to recover from his wounds.

“I would love to see Conan presented a Purple Heart with a live ceremony,” he told The Post. “I think that the K9 is the most valuable asset on the ground. I don’t think that you can replicate through technology or any sort of weaponry what the K9 is capable of … Without a doubt I think that any award — including the medal of honor — should be able to be received by a K9.”

Kit Sawyer’s dog, Roy, was shot by a terrorist in Afghanistan. The 6-and-a-half-year-old Dutch Shepard was the only casualty during a dangerous night mission in Paktika Province. The dog was shot after charging a high-value killer he had sniffed out hiding behind a tree. The terrorist had been lying in wait with an ambush team but Roy’s nose thwarted them.

Roy was treated in a human emergency room at Bagram Airfield, but ultimately succumbed to sepsis. Sawyer and his team were devastated.

“Me and one of my old commanders said if I could, I would submit him for a Bronze Star for valor and definitely a Purple Heart,” Sawyer recalled.

In his final hours, the troops fashioned Roy a laminated, paper Purple Heart that they attached to his collar.