This project is wood refinishing 101.
No amount of Murphy's soap, wiping, washing, smearing or smathering of any product is going to restore the discoloration due to fading/UV damage. You may mask it a little with something with a pigment but that isn't restoring to the original.
The process to refinish the walls isn't a whole lot different than trying to restore a paint job. I am sure many of you have seen a spot on a car paint job where maybe a bumper sticker was removed after a few years? The paint that was protected from the sun/elements is a different color (usually darker) from the rest of the car. To even out the color (without repainting) You remove as little of the top layer of paint as possible (wet sand) to get to the good paint below, on the good and bad surfaces. Then you level the surface with ever finer grits of wet paper and then polish. This is a similar process. Suggesting that cleaning and wiping on some goop is going to fix this issue is like suggesting washing and waxing will fix the paint job.
The objective is to take the least amount of the damaged top wood off to get to good wood. The good wood is identified by the green arrow. A 6" or 8" orbital with 80 (100 if you are more comfortable) will work and make quick (but safe) work of the top layer. It has a flat pad and works at a relative slow speed so no risk of gauging unless you force an edge into the wood. The key is to keep the sander moving at all times. DO NOT stay in one spot. When 95% of the yellow is gone you can switch to the next lower grit. Not to much material is removed at one time, so it will take several passes to achieve a 95% removal at 80 grit with an orbital sander.
You do need to be cautious of one thing, there is a V groove highlighted by the red arrows. You do not want to sand these away, unless you want a flat panel. They appear to be about 1/8" deep so if you do start to sand them away you are sanding WAY to much. Second, bridge the V so to sand it evenly, keep the V in the middle of the pad. You do not want to sand one panel on one side, then the other panel on the other side, to much risk of dipping into the groove and softening the edge.
Your entire sanding start to finish 80/100, 150, 200, 220 should not remove much more material than the nicks in the wood highlighted by the black arrows.
