'Conflict Status' & 'Conflict last run' fields are not getting updated on Change record until after the record is reloaded.
Conflict of timing
We confirmed that its just a timing issue and adding delay will fix this. As long as you don't have any issue with adding a bit of delay then it should be fine.
In testing, we added below code into the ChangeConflictWorker Script Include within the start method.
//adding some delay to try and fix live update not coming when the form loads.
You may want to try some other method of adding this delay or make the process synchronous and the code above is just a way to demonstrate that the issue is caused by a conflict of timing and shouldn't be considered a solution unless the you decide to add that based on your own assessment of the situation.