In my opinion...
The various home remedies, e.g. ice cube sloshing, the
Geo Method, Calgon water softener, etc. all have some potential but none of them are a sure thing. Worth a try, but don't get your hopes real high. The ice cubes probably add a bit of abrasive action to the general sloshing and so may help a bit, but it's more likely the sloshing of mostly clear water that does the trick, assuming it works at all.
I favor a more rigorous treatment using Rid-X or
Roebic K57 and letting it stew in a full tank of water for 3+days before flushing. That's not foolproof either, but it has a solid track record of working, i.e. "digesting" most all the nasty gooey stuff.
Frankly, if a lot of TP is clinging to the common thru-wall type of tank sensor, it's going to be a PITA to get rid of. Some types of TP dimply don't dissolve very well and once it gets draped over the sensor pins it can be extremely difficult to get rid of. It may take several treatments, whether the same one repeated or multiple methods.