Hi, i can see your problem about finding the Firmware. I have found these two links after looking for ages.
It would appear from reading up on this that the 22A drive is not listed on any of the Sony support sites. It also has been suggested that it is a liteon drive badged as a Sony. This is quite common, some Philips drives are actually Ricoh and so on.
The above link opens a strange window about winning a prize, dont be put off, the download will commence for you to save to a drive folder.
The link below provides the tools for doing the upgrade.
I must advise you that I do not have your drive model so cannot vouch for the files but all looks coshe. "LtnFW" and "MKFLASH"
Good luck and post back the results if you go ahead.