Just yesterday, RoxFox64 met up with Hellcat and Davee who were in the final stages of making their new, signed downgrader. As it turned out, he had upgraded to 6.35 from 6.20 recently and needed to use it, imagine that coincidence! After waiting for Davee to give him the compiled version to test, he was more than happy to see that it had worked 100%!
With this downgrader, it will be very simple, and very safe, to downgrade from 6.35 to the 6.20 firmware!

From the ReadMe:

[What does it technically do?]
It uses the utility/power exploit to gain kernel access and reboots into the updater with a special PRX running. This PRX uses the pspdecrypt functionality to decrypt the updater PRX when needed. This allows the updater to boot in the newer firmwares.
So, in order to use this you need the official update for OFW 6.20 (SPECIFIC TO YOUR MODEL)
Installation is very simple:
Place the 6.20 update in X:/PSP/GAME/UPDATE/
and extract the downgrader’s folder into X:/PSP/GAME/

Why OFW 6.20?
Currently, 6.20 is the latest firmware we have full Kernel access on. This is the firmware you’ll find the absolute most functionality on along with HEN TN-C.

Discussion: Here

Downgrader Download: Here
6.20 Official Firmware (1000, 2000, 3000): Here
6.20 Official Firmware (Go!): Here

Total_Noob, GEN team member has apparently finalized what will allow PSP 1000, 2000, 3000, pspgos owners to use their homebrew. There is no current release date to be announced but at least before Christmas. Something to note: This flaw is in PSP firmware 6.30 and 6.31 but is less stable which is why he is concentrating on 6.20.

So, Total_Noob found a way to enable launching homebrew on the PSP via the XMB! If you guys were around the PSP scene to remember ChickHEN or even the HEN from the 2.XX era, then you’ll be familiar with the term HEN and what this kind of stuff this new exploit will allow. Currently, it works up to the 6.2 firmware and launches from HBL (homebrew loader) from the Patapon 2 or the Hot Shots demo from the PSN.

[15/10/2010]
- ParitionCheck has a new structure, fixed (now you can load unpacked prxes), thanks to Davee!
- One patch of ModuleManager was not right, fixed (now you can run all homebrews), thanks to Davee!
- Added some systemctrl exports

Discussion: Here
Source 1: Here
Source 2: Here