Here is the official link:
So you should be covered by the cap-gap even if you are on your grace period during that time. It says, "Which petitions and beneficiaries qualify for a cap gap extension?
H-1B petitions must be timely filed on behalf of an eligible F-1 student. “Timely filed” means that the H-1B petition (indicating change of status rather than consular processing) was filed during the H-1B acceptance period, while the student's authorized duration of status (D/S) admission was still in effect (including any period of time during the academic course of study, any authorized periods of post-completion OPT, and the 60-day departure preparation period, commonly known as the “grace period.”)"
However, you would not automatically be allowed to work:"What if the post-completion OPT expired before April 1? It appears that F-1 status would be extended, but would OPT also be extended?
A student who completed his or her post-completion OPT and who subsequently was in a valid grace period on April 1, would benefit from an automatic extension of his or her D/S admission under the cap gap, if the H-1B petition is filed during the H-1B acceptance period, which begins on April 1. The employment authorization, however, would not be extended automatically, because it already expired and the cap gap does not serve to reinstate or retroactively grant employment authorization."
You can work closely with your Designated School Officer (DSO) in that regards XXXXX XXXXX if they can get you an EAD during that period, but you may not be able to get one.
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