"GTY" is not a special rate, it is essentially the highest tier rate because it comes out after a certain percentage of staterooms have already been booked (possibly with tier-increases between opening day and the GTY). So by default, "GTY" isn't going to be a better rate than opening day. You do not get cabin choice with a GTY but are guaranteed a cabin of that specific level (i.e. 11A, 9B, 7A, etc.) or higher.
What causes confusion is the "*GT" rates -- IGT, OGT and VGT which appear on confirmations noted with stateroom "GTY". It's the *GT rates that are special offers, typically reduced rates with restrictions. These rates may or may not be lower than opening day fares. As with GTY, you do not get cabin choice but will receive something in the broader category of interior, oceaview or verandah.