I suspect that the Gov will veto the bill to protect his political a$$. He has too many gun control toddies he has to appease to roll-over without some illusion of a fight. That will put the ball back in the legislature's court (aka special session) where they will be forced to "put up or shut up". In the end they'll have to override the veto or let "Constitutional Carry"/local ordinances chaos to ensue. I don't think the legislature can permit that. Amy will be forced to seek an 11th hour stay from the 7th circuit since the Gov vetoed the legislature's bill. Assuming the legislature overrides the veto the Gov will then order Amy to file a petition for cert as well as seek a stay while awaiting the SC, again to protect his political a$$.
Assuming Amy is forced to seek an 11th hour stay from the 7th I'm not sure they'll comply. It seems plain to me that the legislature wasted most of the 6 months they were given coming in just under the wire with a bill, but not legislation signed into law. They also had more than adequate time to file a petition for cert that they clearly don't want to file. Had Amy filed a petition it may have given more credibility in seeking a stay at this point from the 7th, but without it I don't think (maybe hope) that the 7th will comply.
This is just about to get interesting. More "extend & pretend" tactics from the gun control crowd.