Is it OK to start new employment after 60 days if H1B petition is filed in time?

[gcshahf2a]

I was laid off mid-April this year. My 60 day grace period ends on June 12. I have another offer, and am planning to start with the company in June. We are in the process of H1B transfer (LCA filed, waiting for approval, after which H1B petition can be filed). Due to the tight schedules and other policies, the company has insisted that I start on June 15, a Monday. I was told by the immigration team that the 60 day grace period clock stops once the H1B petition is filed before June 12, and they have assured me that they will file the petition before June 12, with a few days to spare. Is it true that if the new petition is filed before June 12, I can start on June 15 even if June 15 would be day 63 from my lay off, without affecting my H1B status?

[R78]

The 60 day grace period is to find a job, file H1 and obtain receipt from USCIS
You can start working on June 15