I thought I had until the end of June, and the whole of the 30th to finish my game, but it actually ends 11am in my time zone. (I had been dealing with very bad jetlag for the past few days, so my time management has been off.)

It would be good if the deadline gets extended. If not, I will upload what I have and update my game afterwards.

Yes, no problem. Just PUBLISH your project on itch.io and SUBMIT it to the jam.
Since this is not a ranked jam, you can upload the final version this weekend.