Hey there,
Lots of theories in here...
Generally speaking, valve sticking is a result of oil coking in between the valve guide and valve. Eventually it builds up enough that the valve stem is pushed against the guide hard enough that it will friction weld itself to the guide. This seems to be accelerated by excess heat or excess debris in the oil. Heat cokes the oil faster and dirt increases friction which increases heat. Reducing either of those things does seem to improve the situation. As others have stated,
Lycoming 388 and
Lycoming 1425 give guidance on how to help check and repair it.
Here is a bit more narrative on it.
Lycoming also has done pressure lubricated exhaust valves where it was necessary. Check out the
TIO-540-AF1B in the old eclectic Mooney TLS Bravo. The primary cooling action for valves in Lycomings is the contact cooling through the seat into the head and off the fins, most engines aren't expecting to rely much/at all on oil for cooling of the valve stem.
As far as heat is concerned. Contrary to what many believe, the front cylinders are often the worst cooled cylinders. Airflow coming from the front of the cowl ski jumps over the the front cylinder and ends up cooling the back ones better because it hits the baffling and is forced down through the back fins. Think of the direction the air is flowing into the cowl, it's not straight in. It's at an upwards angle, aimed at the top of the cowling because of the aircraft's AOA. It's also interesting to note that some airframes specifically have higher sticking occurrences than others; which would indicate that it is not an inherent engine design issue and more of a installation/cooling issue. The #2 cylinder has the exhaust side facing back, so if air is skipping over the top that cylinder, that side of the head will be hottest. That's why you likely see it on the #2 and not #1. Lycoming tried to help with this by giving that side of the head significantly more fin area; look down at the head from the top of the engine, the fins are way bigger on the exhaust side. But if air isn't going over that side of the head, it won't matter.
As far as dirt is concerned. I would recommend doing oil analysis (
Lycoming 171). It's not a perfect indicator of engine health but it has good merit. From my experience, if you have a high silicon result, you have a much higher propensity for valve sticking. The silicon is dirt/sand that is making it into your oil somehow. Usually this is via an induction leak somewhere behind the filter; stated differently, your engine is ingesting dirty, unfiltered air somehow. Find and fix those leaks!
For homebuilders. Before you go modifying your engine with something that will affect your oil pressure (don't do that, please); you could have unexpected pressure drop issues as you go up in temperature or up in altitude that you don't see on ground testing. I'd recommend adjusting some things on the baffling. You could try building a vane or a scoop that directs some air down across the exhaust valve side of the cylinder you're experiencing issues with. A lot of the helicopter OEMs do this because their installations tend to have the least amount of available cooling air (fan vs airspeed).
I don't personally believe that oil pressure, as long as it is in the green range, has any significant effect on valve sticking.
Hope that helps.