Under the "Full Faith and Credit" clause of the constitution there should be no question about reciprocity:
Article IV, Section 1:
Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof.
This is why your driver license, registration, marriage, divorce, etc. are universally recognized. There is no reason a concealed firearms permit should not be included!