Need advice on H1B cap-exempt petition while unemployed on STEM OPT

Hi everyone,

I need some advice regarding my current visa situation. Here are the details:

• I'm currently on F1 visa and recently got my STEM OPT extension approved.
• My last day with my current employer is on 5/17/2024, and I will start accruing unemployment days on my STEM OPT after that (I have more than 140 days of unemployment left)
• I have a tentative job offer from a cap-exempt company starting on July 1st, but they don't have an e-verify number, so they plan to sponsor my H1B visa (cap-exempt) and apply for premium processing sometime between May last week and June 1st week. This will be after my employment ends with my current employer and I will be unemployed at the time of filing this petition.
• The company's attorney mentioned there might be an issue with petitioning for a cap-exempt H1B while I'm unemployed on STEM OPT. However, I can only start working there once my H1B is approved and my status changes to H1B.

Given this situation, I'm concerned about the legal implications of being unemployed on STEM OPT while my H1B petition is being processed. Can anyone share their thoughts or experiences?

Thanks in advance for your help.