Quick update:
I installed NE2 bulb on a solid state PS ULII and indeed, the bulb light on for 6 seconds upon startup.
So this solution works great. Thanks TriodeLuvr.
I used to use diodes, but I think I prefer the neon light for nostalgic and visually appealing effect

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While I was there I wanted to check also the same on a tube rectifier ULII. I wasn't sure the online time period of the rectifier would be the same or close to the driver tubes.
The result is... no glow. So all the tubes come online pretty much together so it's an happy family startup.
All right, now I am sure.
I'll retrofit the neon bulbs on the UL120, maybe reducing the stability network caps a bit, if I feel like it as I can't hear the difference honestly.
So I put back the black UL120 on the bench and set back another little black Acrosound critter, the one I always come back to, no matter what.
Gosh that tiny thing beats then all.
