I'm not familiar with this particular HP printer model, but it's interesting that you say this happens only toward the end of the print.
I've had this kind of behavior on other printers caused by flexing of the sheet as it exits the printer. Toward the beginning of the sheet, there's not much paper sticking out past the printhead, so not much flex. Toward the end, when most of the sheet already has been printed, the weight of the paper causes it to bend slightly in the paper path, making it more likely to be scraped. (This may be why your HP tech suggested your paper was too thin: thinner paper is more likely to flex.)
You might check to make sure nothing is obstructing the exit of the paper, and that you have unfolded or extended all the pieces that support the sheet as it is coming out of the printer.