I've had some, generally intermittent, success by working through the gentler solvents I have about the place. Things like neat washing up liquid, isopropyl alcohol, computer screen cleaner, foaming aerosol cleaner (RS breed in my case), sticky label remover, Jizer, Gunk, et al. Apply a little at a time on small areas and rub.
If you can find the right one the results can be good for a year or so but there seems to be no lasting cure. Lots of effort for little reward. 50% success is doing well methinks.
Wear disposable gloves because a seriously wrong cleaner pick will create a super sticky gunge that seriously doesn't want to wash off your fingers.
I have the impression that regular use of foaming cleaner holds back the deterioration can prolong the life of a cleaning session. Letting finger sweat marks build up to the sticky stage seems to be a fatal mistake as the sticky seems to spread from such marks.
Clive