[pxl_rip]

Just to be sure... This time I'm not going to reply and wait directly for their next email?

If I don't get any communication in the next day I'll get back in touch with them?

Email them and say that given they're Investigating It with the carrier, there's no reason for you to do the same.

If they argue that point, tell them that you've purchased and claimed through them, not the carrier, therefore all communications will be between you both.

I've replied with this.
But since I've got no reply how many days should I wait before contacting them again?