Honestly the theory behind this is a bit suspect. There are plenty of turbocharged cars out there with close-ratio gearboxes. You will still be quicker with a close-ratio gearbox with a short final drive. The main benefit of an LS box is that since a turbocharged VTEC motor generally widens its powerband compared to a normally aspirated one, a taller overall gearbox can be used to make cruising more relaxed.

If you already have a GSR transmission, stick with the GSR transmission.