Well, only if you have an R3 Touring. If you have a Roadster, you need a Roadster-specific tune.
I saw that other thread, and the fact that you notice the lack of power in the lower gears seems to indicate that your secondaries are still interfering. Triumph purposely limited 1-3 to make the bike a bit less dangerous to newbies I guess. If you need more power, you have numerous options - first of all, a tune that sets the secondaries to 100%, then optimized for your bike. Having an expert put the stock Roadster on a dyno can squeeze out 20 hp more without even changing air filters, I believe; if memory serves
@Neville Lush did just that and had a thread about it here.
This is because no two engines are alike, and just bunging in an aftermarket custom tune into the ECU will never get you 100% there. The only great way to get the bike optimal is to find an expert with a dyno.
Combined with said expert, new air filters and an aftermarket exhaust can get you to 160-170. A Carpenter kit can get you to 210 or 240 and with that mod your bike will rev to 9000.
It's all about how much you want and how much you want to spend.
Anyway, to just open the secondaries, go to:
Scroll down until you reach the Roadster section. Verify your bike's VIN matches the range they specify. The first map there seems to be for a Roadster with stock exhaust that merely opens the secondaries and raises the speed limiter.