St165 Back-up lights won't stay lit in reverse.

I just noticed my back-up lights are not working for more than 1 sec.
They lit, then go off after 1 sec, although the gear is still in reverse.

Therefore, I decided to perform the following different tests:

1- Remove both back-up light bulbs (thay are new), and using a mutimeter, check for current at the back-up light bulb location. Surprisingly, the multimeter shows a solid 12V reading at the bulb location, for as long as the shift lever is kept in reverse, altough the lights bulb themselves won't stay lit for more than 1s when they are installed back at their proper location. This should usually indicate the back-up light switch is working properly.

2- So I jumped the back-up light switch, located under the air box : the back-up lights stay lit properly, as long as the jump wire keeps letting current through. This indicates the reverse switch itself shoud be the culprit.


So here is my problem : both test conclusions are contradictory.

My personal guess is : It seems like the back-up light switch is capable of letting some current go through, but not enough for the lights to stay lit longer than +/- 1 sec. Test#1 is probably non conclusive because the multimeter won't drain as much current as the actual light bulbs, therefore the test working conditions are different.

Might also be faulty wiring. I noticed the jump wire can get slighty hot in test #2 (+/-120F) after holding the back-up lights lit for more than 1min. Don't know if this is normal. It seems the current passes directly into the back-up light switch, without activating any relay, so this kind of temperature might be normal (120F won't melt anything, especially in the engine bay...).


Any ideas??