H1 visa

I came to USA as a visiting scholar (J1). I got my second Ph.D degree in USA.
When I searched for jobs, the stuff in our univ. career center suggested me deleting my first Ph.D degree (in China) from my resume. I got a position in a univ. using the resume. I am required to provide documents (current resume, degree certificates, etc) for H1 visa application.

Should I add my first Ph.D degree in my current resume and provide my first Ph.D degree certificate? I think there was the record of my first Ph.D degree when my boss applied for J1 for me.

Thanks!!