I’ll have to do some looking as my bike doesn’t have a gear shift indicator. OK, I do have one, but it is aftermarket.
A quick look at the wiring diagram shows a gear position switch that puts a ground on one of 6 wires that go to the meter. Neutral has no wire so when in neutral no wire has a ground on it.
With no wire having a ground you would have a neutral indication at the meter. Since you have a 4 showing then the white wire would normally be grounded. Assuming (yes, I know

) the meter is good, I would suspect the switch.
I don’t know how it is mechanized (connected to the shifter) though. But by disconnecting its connector You could easily check the switch with a multimeter while working the shifter through each gear and metering each pin to ground. I’ll have to look in the manual some to see where it is and how it works with the shifter.
OK, if you have that manual, look on page 19-38. It shows how to test the switch by doing what I said above. Picture makes it easy. If it has to be replaced it doesn’t look too difficult, just a little pain in the, well, you know.