In theory you are correct, in practice the hotel will play dumb and say they'll match it but after you file paperwork proving it was cheaper somewhere else. I've literally booked on HT from the front desk because they can't match the rate. It makes no business sense, but yet here we are.

Another example: I just checked HT for San Francisco tonight and the Hyatt Regency is up for $179 after taxes and fees. On Hyatt.com it's $259.