The battle over the proposed millionaire's tax has been oversimplified as a political death match between Gov. Phil Murphy, its chief champion and Senate President Stephen Sweeney, D-Gloucester, its determined foe.

But what about Assembly Speaker Craig Coughlin? Remember him?

Coughlin, a mild-mannered lawyer from Woodbridge, who carefully parses every sentence, has also made it clear that he has no desire to schedule a vote on the millionaire's tax this year. But for the most part, Coughlin has quietly retreated off-stage as Sweeney and Murphy continue waging their nasty, petty blood feud that has paralyzed state government.

Until now.

Coughlin steadily flexed his political muscle last week by rounding up nearly 10 Assembly Democrats and long-time supporters of raising taxes on millionaires to effectively declare surrender.

They announced that they would not let their ideological devotion to a tax hike get in the way of supporting a fiscal year 2020 budget that isn't propped up by hiking taxes on 18,000 millionaires.

“I will support a responsible state budget even if it does not include a millionaire’s tax, while still being a staunch advocate for the millionaire’s tax until it gets done,'' said Assemblywoman Britnee Timberlake, D-Essex.

There is no other way to see this than as a setback for Murphy. He is already facing the prospect of unified opposition from nearly every lawmaker in South Jersey, the political fief of Sweeney and his power-broker-benefactor George E. Norcross III.

But the Democrats who walked back their support last week come from Hudson, Passaic, Bergen and Essex counties, the North Jersey heartland where Murphy could expect to garner votes. Now, he can't count on them.

For all intents and purposes, they've abandoned the fight. They could have used their collective power, formed a bloc and pressured Coughlin into changing his mind. They chose the opposite and fell in line.

Sweeney:No tax increase for NJ millionaires, no $125 tax credit for anybody

More from Stile:It's a personal fight over NJ tax breaks, legal weed and more but Democrats aren't talking

And in doing so, they left Murphy with a steeper hill to climb. He's heading into the home stretch of the budget season with grassroots activists and public employee unions standing at his side — but they don't sit in the Assembly and cast votes. Nor do they have the support of the most important person in the chamber, Coughlin, who simply could doom the measure by refusing to post it for a vote.

"Supporting something that will never pass is a Pyrrhic victory, but doing so at the expense of New Jersey families is no victory at all,'' explained Assemblyman Gary Schaer of Passaic.

Coughlin's round up of Democrats took place after two Assembly members from Hudson County publicly declared support for the tax hike, according to New Jersey Globe, a political website. It was a minor public relations victory for the governor, according to New Jersey Globe, a political website. Coughlin's staff responded with a nip-it-in-the-bud strategy and by week's end they converted other long-time tax hike devotees into unlikely ones.

In some ways, nobody should be surprised. Coughlin's voice may not rise above the level of a school librarian, but he made his intentions known loud and clear from the outset.

He was not going put his members in the position of voting for a headline-grabbing tax hike in an election year. All 80 seats in the Assembly are up for reelection and sit at the top of the ticket in November. Candidates will not be down-ballot afterthoughts as in most elections, but prime targets for attack.

Despite polls showing strong public support for taxing millionaires, no amount of cajoling, or sweetening with the promise of a paltry $125 tax credit, or shaming from class-warfare themed television ads produced by New Direction New Jersey, the dark money group run by Murphy operatives, was going to change Coughlin's mind.

Just last week, the New Direction turned up the heat with a digital campaign, implying that Coughlin and Sweeney were toadies of Donald Trump who coddle the rich at the expense of the middle class. Coughlin's staff effectively replied with its corral of pro-tax hike defectors.

Assemblyman John McKeon, of West Orange, has been a long-time champion of raising the rate on those who earn $1 million or more income from 8.97 percent to 10.75 percent. The hike would raise an estimated $450 million in revenue for schools, property tax relief and other Murphy "fairer and stronger" reforms.

Yet, McKeon also refused to take an unbending "line in the sand" demand for the tax hike.

An unbending demand for a tax hike "doesn't foster the environment for a compromise,'' McKeon said. He cited, for example, of alternatives that Murphy could offer in negotiations in the next few weeks — such as a proposal for a smaller millionaire tax hike or one that is phased in over a few years.

Murphy still has some cards to play. He is still a governor with enormous advantages to coerce members to change their mind — stalling promised patronage, closing off state agencies to a legislator's powerful allies and donors, or vetoing cherished bills. He would have to play the kind of Trenton hardball he has largely avoided.

He could also continue to build public pressure with the New Direction media campaign, financed with at least $2.5 million from the New Jersey Education Association.

Maybe the grassroots fury in that propelled a Democratic Party rout in last November's midterm congressional races will unleash its anger on Trenton. Members of the South Orange-Maplewood Action, a group that mobilized Congresswoman Mikie Sherrill's successful victory in the 11th Congressional District last year, quickly launched a Facebook effort Friday urging members to call McKeon's district office and demand his support for the tax.

These are cards Murphy and his allies will play, but they are wild cards. Sweeney has said repeatedly that he won't post the tax unless Murphy signs on to his "Path to Progress" cost-cutting measures that includes the state worker givebacks. Murphy, who is closely tied to the unions, has no desire to embrace Sweeney's plans.

And even before Sweeney could take up a millionaire's hike, it first must get passed by Coughlin and the Assembly, where all revenue bills must originate. Coughlin isn't budging.

Coughlin didn't have much to say last week — he let the one-time ardent tax hike supporters do the talking for him.