(CNN)Former President Donald Trump on Tuesday backed Alaska GOP Gov. Mike Dunleavy for reelection in 2022, but his endorsement came with a caveat that is just the latest sign of his attempts to influence Republican politics next year.

       Trump vowed to rescind his support if Dunleavy endorses Republican Alaska Sen. Lisa Murkowski, whom Trump has made a top target in next year's midterm elections after the senator voted to convict him in his impeachment trial earlier this year.

       "Alaska needs Mike Dunleavy as Governor now more than ever. He has my Complete and Total Endorsement but, this endorsement is subject to his non-endorsement of Senator Lisa Murkowski who has been very bad for Alaska," Trump said in a statement. "In other words, if Mike endorses her, which is his prerogative, my endorsement of him is null and void, and of no further force or effect!"

       Trump has endorsed Kelly Tshibaka, Alaska's former commissioner of administration, to take on Murkowski in an intraparty fight that's emerged as a proxy battle between the former President, whose political advisers have joined Tshibaka's campaign, and Senate Minority Leader Mitch McConnell, a Kentucky Republican who is backing the senator.

       Murkowski was one of one of seven Republican senators -- and the only one running for reelection in 2022 -- who broke party lines by joining Democrats in voting to convict Trump for inciting the insurrection at the US Capitol on January 6.

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       In a sign of how many state parties have fallen in line behind Trump, the Alaska GOP censured Murkowski for her conviction vote, which Tshibaka has seized on.

       But Murkowski, who waged a successful write-in campaign in 2010 after losing the GOP primary, fired back when announcing her intention to run for reelection in November. "In this election, lower 48 outsiders are going to try to grab Alaska's Senate seat for their partisan agendas. They don't understand our state and frankly, they couldn't care less about your future. My commitment to Alaska is the same one I made during my first campaign."

       Murkowski could benefit in 2022 from a new system in the state where candidates run in a nonpartisan primary and the top four finishers advance to the general election. The winner is then decided by ranked-choice voting.

       Dunleavy won the governor's race in 2018 and is running for a second term next year.

       "Mike Dunleavy has been a strong and consistent Conservative since his time in the Alaska State Senate. I was proud to endorse his first run for Governor, and I am proud to support his reelection, too," Trump said in his Tuesday statement. "From his handling of the virus, support of the Constitution—including the Second and Tenth Amendments—taking advantage of all the opportunities Alaska has to offer, and his strong pushback against the Liberal Biden Administration's attempt to hurt our great Country."

       Manu Raju and Alex Rogers contributed to this report.