As far as I know, this is not true.
First things first. Your current employer is processing your GC so that they could employ you in the future. Your current employment has nothing to do with it. As long as your current employer has no problem in keeping the process alive (they finished all their parts anyway), you can work for a different employer now. Of course, if you go by the spirit of the GC, you need to come back to your current employer once you get your GC and if they have a job open for you. I personally have seen a couple of my friends leaving their employers while their GC was under processing.
Now... since your GC process is not at least one year old (Clue: April 2006 PD on labor), I assume you didn't use up all your regular 6 years on H1-B.
Talk to your current employer if they are willing to keep the GC process open. While filing for 485, most of the work is on your side. Your employer needs to do very minimal work, like providing a copy of I140 approval notice etc.
If they agree to it, I think you are in good shape. Just look for another job, apply for H1 transfer and start the new job.
Good Luck.
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Disclaimer: I am not an attorney. I don't know all the laws that exist. Whatever I said is only an opinion based on what I know.