The two series-connected neon lamps form a gas discharge tube protecting the CRT and driver from any static or arc discharges.
They go directly across the cathode-to-control grid of the CRT. Nothing to do with the trigger section. The two clamp diodes D407, D408 prevent any +ve CRT grid bias and the neons limit -ve grid bias to ~180VDC.
They should never be glowing except for maybe a blip on power up/down.