912 ULS randomly dies at idle

Just start with a basic diagnostic and rule out some easy things to check. Do a few easy and cheap checks first before thinking it's some exotic problem. This just rules out some common issues that won't cost you any money. 

Since you already pulled the carb bowls I take it they were clean. Make sure the throttle and choke levers are opening and closing properly. Weigh the floats and make sure one or two aren't so heavy it's just flooding. This shouldn't be an issue at the 3500 rpm, but since it quit at low rpms and you're already in the carbs might as well clean the idle jets to make sure it isn't a contributor. Unscrew the idle jet and clean it out. The idle jet is in the recessed hole next to the main jet. Use a straight tip screwdriver to unscrew it. It should have a tiny open hole down the center. Next make sure your plugs are all gapped properly if you're using NGK's and no thermal paste is on any electrode tip. Then unscrew each plug cap and trim the plug wire back about 3/8" to get some good clean wire. Apply a dab of dielectric grease on the wire tip and screw the cap all the way back on until it stops. If you have a new set of plugs lying around add replacing them to your list. Won't hurt. After these couple of quick, cheap and easy checks put the carbs on a set of gauges and check their sync and the gauges will tell you if it's a fuel issue. It may be a plug and or wire issue. .

Roger,  Thanks for all the great tips but it turns out the solution was much simpler than all that.  I discovered today that I had the idle set at about 1500 rpm and it was right on the ragged edge of killing the engine.  When I reset it to 2000 rpm the issue went away.  However, just as a point of interest, in your experience what is the lowest rpm that the 912 ULS should be able to operate at consistently without killing?

The Rotax 912 operators manual says 1400 RPM is the "minimum idle speed".  Neither the Light nor the Heavy maintenance manuals specify an actual idle speed.  They both refer to "checking" idle speed. 

But I've always heard idle should be set at 2000 rpm, except for seaplanes, where it can be set at 1800 rpm to facilitate water handling.  (Do NOT take my word for that - I am NOT an expert on Rotax engines.)  

Update:  Just re-watched the "Carb Synch 4" video here on this site, and it recommends idle RPM be set at 1800-2200 RPM, so I guess that's where I "heard" it before...

" I discovered today that I had the idle set at about 1500 rpm and it was right on the ragged edge of killing the engine."

That would do it. Easy fix. I like easy. :) 

Remember when you look at the maint. manual it is written for more than one engine. Make sure when you read and look at engine info you're looking at the info for your engine. (912UL, 912ULS or the 914) The UL has a lower compression than the ULS and a little lower rpm isn't as critical. That said you're still better off with a higher idle rpm that YOU are controlling. The higher rpm allows for the engine and gearbox to be smoother and less compression impulses vs low rpm idle. Setting the idle at 1400 for the ULS may cause it to stop on approach and beats the gearbox up. I set idle at 1700 +/- with the understanding with the owner NOT to sit there and just idle. It beats things up. Vans RV12 company sets theirs to 1650. I prefer not to go that low. So long as you know not to sit and idle too low and that 1700 ground static rpm is okay for approach because the prop will windmill faster and not right at 1700 and it's okay to pull back to 1700 when you go to turn the engine off, but NOT to just sit there and let it run.

So if you just have to sit and wait for other aircraft to clear a runway or just talking to ground on the radio try running around 2000 rpm.

The whole idea behind not idling too low is to save the gearbox until TBO and not have to replace the gear set inside because you let it sit and idle too low for many hours and now it cost you a lot of money to replace.

Roger,  Thanks for your good input so far.  I'm wondering if the 400-500 rpm drop I'm seeing on one side of my mag checks may be impacting my ability to idle back further and stay running?  What should I look for to resolve the issue?