Will an unlisted second job show up on a background check?

I’m currently working two jobs (let's call them J1 and J2), but I only listed J1 on my resume and job application since it’s my primary role. J2 is a secondary job that I haven’t mentioned anywhere publicly — not on my LinkedIn, resume, or application.

I recently received an offer from a new employer (J3), and they’re running a background check through First Advantage. I plan to resign from both J1 and J2 before starting the new role.

I’m a little concerned because J2 might report employment data to The Work Number (TWN), and I’m wondering if it could show up in the background check even though I didn’t disclose it.

If J2 shows up in TWN, will that cause a problem or lead to a failed background check? Has anyone had experience with this kind of situation — especially with First Advantage?