Traveling back from Mexico City with expired H1b Stamp, Valid I 797 and Visa appointment on July 1st



I have a valid I 797 till July 2023 and Expired H1b Stamp on the passport. I had an appointment at the US Embassy in Mexico City for May 20th. During my Biometrics, they said that due to some processing issues, they postponed my appointment till July 1st. Can I use AVR to come back to US in the mean time? The reason I am asking is because one of requirements is that you should not have an application for h1b visa pending. I am not sure if this is only for people that had denial and 221(g) or it applies to anyone with the appointment.

If I try using AVR, will the CBP Officer know that I have a pending Visa Application?
If they don’t know, Can I tell them I am here for a business reason (we have a sales office in mexico)

Also, is there a preferred route - Air Vs Border Crossing to use AVR? I heard going through border crossing they usually dont check as much due to the heavy inflow.

Please let me know

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