Just checked, and can confirm clock illumination does dim when you switch on the side lights.
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It may just be a coincidence, but I've not heard of anyone using canabus leds for the cluster before, and wonder if the additional resistor in these is an issue.
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It shouldn't in theory make any difference as these are just supposed to create the same resistance as a traditional bulb.
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I've always used straight smd leds like these and never had this sort of problem:
SMD LED Link
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I still think the best option is to try going back to a conventional lamp in at least one of the cluster lamps to see if that works OK, or put a meter on the lamp supply on the cluster to see if you've a volt drop problem.
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If the conventional lamp works OK, or you have 12v on the cluster track where the lamp holder makes contact, it's either the leds or a poor connection between the lamp holder and the cluster, or a poor connection between the lamp holder and the led.
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If it's no different with a conventional lamp or your getting substancially less than 12v, you could make sure the multipin plugs are all making a good contact and there is no sign of damage to the tracks feeding the cluster leds.
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After that things get a bit more complicated, as you need to check the supply to the cluster.
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Have you tried turning on he side lights then switching on the ignition to see if the warning lights in the cluster are also dimmer than they should be if the side lights are on, as that might help to narrow down the issue.
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Scrub that.
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I've just taken a look at your video again, and as far as I can see the warning lights are working OK.
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Paul