Im just going to list my thoughts and what I would check just in case you haven't done any of this.
Easiest thing to do is remove the tailwheel fork, clean the fork shaft bearing surfaces, check for burrs, and re-grease.. lightly. The edges of the bearing surfaces can get burrs that I can imaging might cause sticking.
On our -8 we swapped to a DJM tailwheel with the flat profile, also switched to the Screamin Eagle fork from Fly Boy Accessories. This noticeably improved tailwheel handling. And I do mean noticeably improved not just I tell myself it improved because I spent money on it. It's in my list of top upgrades. The Bell fork is a good option as well.
Add a bit of slack to your chains if they are tight. Some play will make everything less touchy and I think it definitely helps in crosswind landings. Allows for the rudder to have a chance to do some work before the tailwheel kicks in. More of a mushy response but I think sometimes that's a good thing.
Check toe of main gear, shim axles if necessary. Not that hard to do, and getting rid of a toe in condition will make a big difference in ground handling.
Check for excessive drag on the main wheels. If I jack mine up they spin with just a slight bit of drag so they don't spin in the air. Don't know if this is an exact science, I just felt it and guessed. Also could be a bearing problem as the post before me said. I haven't heard of that but now I'll add it to my list of **** to worry about.
Check under your wheel pant, is it smacking the ground when the tail comes down and pulling you to the side? Does it not have enough clearance all the way around and causing heavy drag when the rubber is in compression? Fill your tanks and get someone to sit in the plane and check pant clearance all the way around. I like to make sure I can get the tip of my index finger in there all the way around. I know more clearance is more drag, but I wanted it here. If I ever land with a flat tire I want more than less because I know that's going to be a wild ride.
If wheel landing, let the tail come down on its own when possible. Of course there will be those tense landings when you have to get the tail down sooner to hit the brakes, but otherwise ease her down
I can't see tail spring alignment in the horizontal direction causing this issue. Mine isn't perfectly straight. Maybe if the shaft of the fork is canted left or right, I'd check for that. If it is you could buy an undrilled tail spring from some place like flyboyaccessories.com and drill a new one to square it up. Also make sure the fork shaft angles down and back like the plans say (fork bearing shaft tilts forward slightly). Not sure how to change that if its off but the plans tell you to check for it so I guess it's possible, just not sure how that would happen.
Overall based on your description and the fact that it only happens 10% of the time, I would say the problem is with the tail wheel gear spring, the fork assembly, or the chains. Wheel should definitely not rock side to side. Fix that first. Get new parts if you have to and remove that play. Also try the DJM lightwheight tailwheel. The flat rubber instead of the round for some reason improved my handling a good bit. The flyboy fork is definite upgrade as well. So make sure your wheel cant rock, grab the fork and vertical shaft and twist it side to side and check for play. Second.. clean and grease the vertical fork bearing surfaces and check for burrs. Your wheel could be sticking to the side just right to start the right turn. Third, and I just thought of this one and it may be a good one, check your chain clevis's.
Add safety wire to them so they can't spread or flip upright and jam in the rudder attach holes or the control arm. This used to happen on mine because my chains have some play (by choice).
It could be happening with yours and making one of your chains tighter than the other and holding the wheel sideways. Lastly, pay attention to your feet. You could be biasing the right side pedal or brake and not even know it. I'm sure youve checked for operator error, but I know someone who used to do this alot and deny it every time
Let me know what you find out.
-Matt Jr.