Members of Welsh Labour will decide whether or not to back plans to increase the number of Senedd members and change the voting system for choosing them.
They'll discuss the proposals, which have been put forward by the Welsh Government in partnership with Plaid Cymru.
Delegates are thought to likely back the changes although there is some internal opposition with a number of local parties voting to reject them.
If Saturday's (July 2) special conference votes against them, it could scupper the plans altogether.
The Welsh Conservatives have called for a referendum before any expansion of the Senedd takes place, asking why Labour members get to vote but the wider public does not.
These proposals would see the number of members increase from the current 60 to 96.
The way they're elected would change drastically.
Gone would be the way they're chosen now, which is by a mixture of traditional ‘first-past-the-post’ contests in constituencies and an additional 20 members chosen for regions through a system which is intended to reflect parties' actual share of the vote.
In its place would be 16 mini-regions created by combining pairs of constituencies based on the expected new Westminster boundaries.
Voters in each region would choose six members from lists selected by the political parties that would be legally required to meet gender quotas.
The planned changes have been backed in a vote by Senedd members. Plaid Cymru party members have previously supported the party leadership's position and Welsh Labour members voted at their conference to back Senedd reform.
However, they'll decide today (July 2) if they agree with the final specifics of the plan.
If they say 'no', Mark Drakeford will have to go back to the drawing board.
He’s thought likely to be able to win the support of a majority of his members but it’s by no means a certainty.
Two local Labour branches (CLPs) have voted to reject the proposals and it's thought others are also opposed.
If accepted, the changes will be introduced in time for the next Senedd election in 2026.
Supporters of reform have long argued that there are not enough MSs to scrutinise Welsh Government properly, once government ministers and presiding officers are taken out of the equation.
The Welsh Conservatives are opposed.
Andrew RT Davies described the plan as a stitch-up between the two parties and says such a big change should be subject to a referendum.
"This is typical pork barrel politics from Labour and Plaid here in Wales," he said.
"Ultimately Mark Drakeford and Adam Price delivered a diktat to the committee that was looking at this.
"We don't agree with additional members for the Senedd, but we engaged with that committee over the voting system and the constituencies that might be required to enhance the Senedd's ability to take on these additional members.
"And so we think it is a typical Labour-Plaid stitch up, pork barrel politics at its very worst. That's why it should go to a referendum and let the people decide."
The leader of the Welsh Liberal Democrats, Jane Dodds, said that she hopes that "Welsh Labour delegates do the right thing and support a much fairer and more democratic voting system in Wales.
"Too often Labour have blocked electoral reform across the UK and now is a chance to put Wales’ interest over their party political interests.:
These proposals come as part of the cooperation agreement reached between Labour and Plaid Cymru, who are working together to deliver some aspects of Welsh Government although Plaid has not entered into a coalition government.
It is not yet clear how much the change would cost.
The Senedd Commission, which runs the Welsh parliament, estimates that an extra 30 members would cost between £11.7m and £12.9m.
That figure is not only for members' salaries, it would also include pay for Senedd staff, running of elections and the upkeep of the parliament building and associated offices.