Say you can work for any employer without sponsorship
This happened to me 6 years ago with the company I’m working. During the initial interview I said vaguely that I had a work permit however she didn’t catch it, fast forward a few months later I was filling out the I9 and I pulled out my EAD and she said “you didn’t specify you needed sponsorship”, I told her firmly I didn’t need sponsorship and that I could work for any employer so she didn’t say anything else. I sounded a bit mad but you need to be firm because HR believes they know about visas and the reality is they know very little. I’m sure some HR ate more knowledgeable than others but my experience tells me they know very little about statuses and visas.