CapGap employer vs Sponsored employer

[Mike Chris]

Does my Cap Gap employer need to be the employer who filed h1b petition for me?

In other words, my current employer didn’t sponsor me H1b while future employer X has sponsored and petition is selected in lottery and my work authorization is extended till Sep30 as per cap gap rule. I wanted to move to employer X once the petition is approved, but can I use the cap gap work authorization to work for current employer? Or would it be valid only if you stick with sponsored employer?