I have a 1992 ZXR400 L2 which I had a similar problem to back in 2014. The problem was definitely fusebox related, as a new fusebox solved the issue, BUT... it must be a 1080 fusebox!
My bike has no off/sidelight/dipped switch on the RHS. It also has no sidelight in the headlamp unit. When you switch the ignition on, the tail light comes on but the headlight doesn't. It is only when the engine is cranked and starts that the headlight comes on.
When it didn't, I found a post about a similar issue on another Grey Import 400 (Honda VFR400? CBR400?) which cited the relays in the fusebox, so I thought I'd give it a try. It worked... so I assume this is a popular Japanese market spec which the UK models, with their headlamp switch, don't require.
A friend lent me a UK fusebox (a 1072 if I remember correctly?). Worked perfectly, except no headlamps... I didn't open it up, but assume it either lacks the relay, or the PCB is different and has that relay on a redundant circuit.
In summary, I'm certain the issue is one of the 2 relays in the fusebox, and I am currently looking at getting new relays to solder into the PCB board to see if I can repair mine so that I'll have a spare.