Minimum recommended time between two marathons?

I just completed my second marathon and had a personal best time of 4:37, which was a 22 minute improvement over my first race last year.

I had planned on focusing on improving my 10k and half times before doing another full marathon, but I noticed there’s a very convenient marathon for me (the Oakland marathon) just five weeks from now. 

I’ll definitely be taking appropriate recovery time before my next run, and I know I can’t meaningfully add fitness by then. But, I suspect environmental factors contributed to me hitting the wall (heat was more than I’m used to, and a 3:30am wakeup time).

Is it feasible to make another personal best attempt so soon? I was aiming for sub 4:30 with this last one. Any input is very welcome!