It's holiday season - lots of partners/heads of departments will be on holiday with their kids, some probably wont be back until next week where a lot of the private schools do not go back until Monday.
So give it a couple more weeks. 10 days seems like a long time to you, to a partner/head of department, it feels like 10 hours.
If you haven't heard back by mid September, drop a very quick/brief email asking if there is any further information you can provide. If you don't hear back after that, then unfortunately it is just one of those things that is likely to come to nothing.