FRANKFORT, Ky. — Kentucky Gov. Matt Bevin’s plan to overhaul the retirement system for public school teachers would cost taxpayers an extra $4.4 billion over the next 20 years, according to a new analysis, a whopping cost that potentially complicates the proposal’s prospects in the House as GOP leaders struggle to find votes for it.

The analysis by Cavanaugh Macdonald Consulting is the first official scoring of Bevin’s proposal, which he unveiled last month. A similar analysis will be completed for the retirement system that covers state workers.

If Bevin’s proposal is approved, the deficit in the teachers’ retirement system is projected to be $11 billion by 2038 and the plan would be 71 percent funded. If lawmakers don’t pass the bill and fully fund the system every year — something they have not done over the past decade — the deficit is projected to be $9.6 billion by 2038 and the plan would be 80 percent funded.

Bevin’s proposed bill would require the legislature to fully fund the pension system every year. But lawmakers could suspend that requirement in future legislative sessions.

Bevin’s plan has taxpayers paying more in the beginning. Savings would not begin until 2034. The payments would eventually decrease, but the changes would require lawmakers to find an extra $600 million to put into the system in 2020. That would be difficult, since Kentucky finished the most recent budget year with a deficit and is projected to have another deficit this year.

The consultants only projected for the next 20 years, as they are required to by law. Bevin spokeswoman Amanda Stamper said the plan will show “significantly better funding” over 30 years. And she said the state will be able to afford “additional payments over the next 30 years, thereby saving the teachers’ pensions.”

“Governor Bevin put a record amount of money into the current budget for teachers’ pensions and we have always said we will need to allocate more money towards all the plans,” Bevin spokeswoman Amanda Stamper said.

Bevin’s proposal would eventually close the pension system and replace it with a 401(k)-style plan. Current workers would have their benefits frozen once they reach 27 years of service, after which they would be moved into a 401(k)-type plan or the remainder of their career.

Bevin’s proposal would suspend cost-of-living raises for retired teachers for the first five years. That would save taxpayers $2.6 billion in the first year alone. But the analysis found those savings would be offset by a wave of teacher retirements and an assumption the state would not make as much money on its investments as it has in the past.

Because the pension system would be closed to new members, the consultants assumed the state would have to make more conservative investments. That’s why they based their projections on an average 6 percent return on investments instead of 7.5 percent, which is the current assumption.

Stamper, the governor’s spokeswoman, said that makes it “not a fair comparison.” She said the consultants should have provided numbers using a 7.5 percent return and then recommended the state switch to a 6 percent assumption.

Bevin has said he would call a special legislative session to pass the bill, but he hasn’t yet. House Republican leaders have said they lack the votes to pass it. Republicans are still recovering from a sexual harassment scandal that led to the resignation of House Speaker Jeff Hoover on Sunday.

House Majority spokeswoman Daisy Olivo said Republican lawmakers are still reviewing the analysis and don’t have “a final opinion on the results.”

Kentucky has one of the worst-funded public pension plans in the country. The state is at least $33 billion short of the money required to pay retirement benefits across its eight separate pension plans over the next 30 years.

The consultants noted the analysis is a projection, so “actual results can be expected to differ.”