Court-ordered mediation sessions between Gov. Mark Dayton and GOP legislative leaders are set to begin Thursday, but details of the expensive, high-stakes process are largely being kept private.

Dayton’s office issued a statement Wednesday that the sessions with mediator Rick Solum, a retired judge and trial lawyer, will be held Thursday and Friday at the Dorsey & Whitney law firm in downtown Minneapolis, where Solum once worked. The two sides will split the cost of the sessions, which are being billed at $257 per hour, half of Solum’s usual rate.

While it’s unclear how long they could go on, the sessions between the state’s top elected officials are certain to add to a growing legal bill — one ultimately being footed by taxpayers — that’s being racked up in the fight between the DFL governor and Republican legislative leaders.

In addition to Solum, attorneys for Dayton and the Legislature are expected to participate. That means the sessions will cost taxpayers at least $1,088.25 per hour in fees to Solum and the other two attorneys.

The governor’s office said the “private and neutral location” of the talks, which were ordered by the Minnesota Supreme Court, were chosen “to minimize the difficulties associated with inquiry while the mediation is in process.” The statement said neither side involved in the dispute will comment on the process “prior to the mediation being concluded.”

At a news conference Wednesday, Dayton declined to answer questions about the process, citing the agreement on both sides to stay silent.

The sessions are the latest chapter in a dispute that dates to May, when Dayton squared off against House Speaker Kurt Daudt and Senate Majority Leader Paul Gazelka over a tax-cut plan and policy provisions included in the state’s next budget. Dayton ended up vetoing the budget for the state House and Senate, hoping to bring GOP leaders back to the negotiating table.

Instead the Legislature sued, arguing that Dayton had violated the state Constitution.

A lower-court judge sided with the Legislature, but Dayton appealed the case to the Minnesota Supreme Court. The high court found that Dayton’s veto was constitutional, but declined to formally settle the matter until the two sides attempted to work out their differences. A progress report on those attempts is due to the court by the end of the month.

Through the end of August, the governor’s office had been billed $245,000 in legal fees. The Legislature will be billed once the dispute has concluded.