Need Urgent Help with Hiring Non-US Citizen

We have interviewed for an entry-level research position, and our top candidate revealed in interview that she is not a US citizen but could provide paperwork stating her eligibility to work for 12 months, which could be confirmed on E-Verify (which we do not currently use).
She believes after this point in time she would be eligible for an H1B visa based on her skills and her having worked in the position for one year. She is proposing that she could apply for this on her own.
I have little experience in this area, but my notes show that the employer must sponsor and pay for an H1B visa, and I have seen posts to that effect here. We do not do this for junior positions, and I don't see how we could make a case at this point that there are no other U.S candidates who could do this work, even after she's been in the job for a year.
But, my question is what should I do next. I think we either have to make her an offer contingent on her proving her eligibility, or not make her an offer and move to our next best candidate, correct? Can we ask her about her visa status, etc., before we make an offer? My reading says no. Can we turn her down at this point (before having confirmed her eligibility to work) on the basis that while she might be able to work for 12 months, we are looking for someone longer term and we are not in a position to sponsor someone for an H1B visa?
Any guidance appreciated. Thanks.
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