ST. PAUL  — The Minnesota Senate’s top Democrat shot down Republicans’ plan on Thursday to protect their narrow majority, leaving little hope of avoiding a messy fight over who will become Minnesota’s next lieutenant governor and how that will impact control of the Senate.

Republican state Sen. Michelle Fischbach is set to become Gov. Mark Dayton’s second-in-command next week, after current Lt. Gov. Tina Smith is sworn in to replace U.S. Sen. Al Franken. Franken announced his resignation amid allegations of sexual misconduct.

State law requires Fischbach, as Senate president, to become the lieutenant governor if the position is vacated. But her leaving the Minnesota Senate could tip the party balance to an equal number of Democrats and Republicans, at least temporarily.

Fischbach wants to retain her Senate seat while serving as lieutenant governor, which has triggered outrage from Democrats. The Attorney General’s office has said she could be required to leave the Senate.

On Thursday, Republican Senate Majority Leader Paul Gazelka pushed for a special session to appoint a Democrat to take the job instead. Senate Minority Leader Tom Bakk told Minnesota Public Radio News  that Democrats wouldn’t agree to a special session — and Dayton says he won’t call a special session unless all party leaders buy in.