"looks" like a t34 to me, certainly by looking at the compressor side

FYI: a T4 was a standard fit for rs500's
a t34 was fitted to the large turbo escort cosworths, but the are a popular upgrade from standard fit t3's on sapphire cosworths

t34's are common for stage 3

if you are running stage 3 at 360, its more likely to be a t34.

it could well be a t4 which is overkill for 360hp in my opinion, but if it is, that might explain the lag