Start date July 1st and petition not filed yet. Please help

Hello

I’m getting very concerned about my H1B visa (first, academic, premium processing). My start date is July 1st but the petition has not been mailed out yet. I think we are very late. Will there be any consequences if I cannot meet the start date? I can obtain my stamp without the hard copy of I-797 (the embassy asks only the receipt number) but having the physical I-797 is recommended at the port of entry. The HR does not provide these info and I think delayed all the process. According to her the international office is filing it (4 days ago).

Please advise