This contradicts what I have received from reliable sources... the gun club I belong to, a SCPD officer who is knowledgeable on the topic. The gun shop owner where I buy my pistols also told me the same thing with an ATF agent standing there nodding her head.
I know how your feeling.. I have several friends who are in LEO in Upstate NYS, none of which really knew the difference between the permits I had. I could never get a straight answer. A week before I left NY my truck was broken into in Albany. When the officer arrived he immediatly started going through my truck (i didnt think he was gonna search like a drug bust), anyway I told him about the Glock in my center console and had my permit in hand, he read it several times, looked at me with a blank stare and said "does this mean you can carry a gun on you?" -- NO KIDDING
The way I see it, if it became a question, they would check with the issuer, your county judge. Hence playing it safe when not having an unrestricted.