There is very little mention of GJ77A carburetion and upgrades, so here is some information on this topic.
Several different Suzuki air filters have 13780-05A00 embossed on the bottom, but the GSF250V part number is actually 13780-11DA0, as identified by "11DA" printed in white text on the black plastic lid.
In the 11DA0 and some other cylindrical Suzuki air filters, the black plastic lid extends as a sleeve part-way down inside the chamber, reducing the exposed filter surface, e.g. compare
13780-01D00 and
13780-01D50 for the GS500.
Anybody reluctant to use the K&N SU-5589 has an excellent upgrade choice in the unrestricted 13780-01D00, which is also cheaper than the restricted versions. The larger exposed surface provides more air flow, while the OEM filter material provides the same filtration efficacy. The longer
13780-42F00 for the GSX1400 apparently offers even more air flow and it should fit in the GSF air box.
The upgrade might benefit from some carb tuning, but it had no adverse effect in my experience, with fresh carb seals, stock jets and pilot screws at 1.5 turns. If necessary, it can be easily reversed by placing some cotton wool in the chamber.