Emailing after position closes

Hello!

I applied to position 5/21 and it closed 5/29 on the job description it stated anticipated start date was 6/8 there was only 5 business days between closing and anticipated start date. To me this very well could mean they already had someone internally or whatever.

Should I still send an email asking for a potential timeline for the position or just give up.

Should I email the person the position reports to or should I just email the main HR email.