Won't be a problem for LPR.
Global entry approval can be ridiculously strict. My wife and I were denied Global Entry because my wife, who had already become a citizen, had tried to buy a watch on ebay and it turned out it was an unapproved knock off being sent from China or somewhere in that area. She didn't know until we received a letter from customs in Anchorage. We refused delivery and thought that was the end of it. A couple of years later we applied for the Global Entry and at the interview I was asked about buying a watch from China. I replied that I had not (I rarely wear a watch) and was asked the same question a second time. That is when I remembered the issue with my wife. We explained the problem, but since we were accused of importing unapproved items we were denied.
I have found, at least for us, our returns from numerous international trips were quick and no problem even without Global entry. We just went with TSA precheck alone for that benefit getting through security.