There should not be any problem whatsoever.

When we travelled, good folks at Kathmandu Airport Immigration did not know what the Advanced Parole document is.  So, we were held back for some 15 minutes until the senior haakim gave a go ahead. 

Actually the senior haakim did not know what the Advance Parole document is either.  We were getting tired explaining to everybody who came to us scratching his head, "khoi yo kaagaz ta ke ho ke ho?  tapai-ko visa chai khoi ni"

"Yo chaahi Amrika ko Green Card paunu aghi diney Travel Permit ho hajur.  Yehi nai hamro Amrikan visa ho"

"Khoi, haamley ta yesto kaagaz kahiley dekhya chainau. Thulo haakim ko hukkum chaahincha."

So, finally thulo haakim said, "Homeland Security ko chinha cha.  Thik cha hola.  Janu deu."

We were not asked about H1 work permit.

Over at JFK airport, we were in the same line as all foreigners.  There was no separate line for AP holders.  When we got to the immigration officer, he took all the documents and asked us to take a seat at the waiting area.  After about an hour or so, we were given our stamped documents, no questions.  Singapore Airlines personnel brought our baggage to us while we were waiting (very nice of them). 

No problem.

But then, I am not the famous Indian actor to raise a big fuss for being asked to wait an hour.