Can an unbalanced-to-balanced adaptor damage an amplifier?

I just purchased an amplifier that has a 4.4mm balanced (Pentacon) headphone jack.  My Audeze headphone's cables terminate in unbalanced 1/4-inch and minipin plugs.

I don't want to replace my expensive 5-6m cables, so my best option seems to be to pick up some adaptors.  Audeze, however, tells me that using an adaptor to connect an unbalanced plug to an amplifier's balanced jack can damage the amp.

I'm sure that Audeze knows what it's talking about, but I see that the amp's manufacturer (T+A) sells its own quarter-inch-to-Pentacon adaptors, presumably for amplifiers like the one I just bought.  So I want to confirm.

I know that there are two different ways to wire XLR connectors, so I wonder if that's the real issue.

Any opinions?