I believe even with the visa it’s still up to the immigration agent. I came close to trouble once when asked for my H1B visa petition document (not the visa in

I believe even with the visa it’s still up to the immigration agent. I came close to trouble once when asked for my H1B visa petition document (not the visa in the passport). I had a photocopy and was told that wasn’t enough and although they’d let me in this time they expected to see the original in future. I also travelled with a letter from my employer explaining where I worked, job title etc as extra documentation just in case to derisk further.