I got your PM, I'm answering your question here so everyone can benefit from it, and so that I'm not answering it again and again in PM. If all you are doing is cleaning the controls, then what I recommend depends on how far apart you are planning on taking the boombox apart. If the slider control board stays mounted, then a tiny bit of deoxit, worked to clean off oxidation first, followed by a small burst of faderlube is all you need to do. Some amount of residue will remain inside.
If you want to clean off as much of the built up residue and grit inside of the control as possible, and you can remove the slider board from the boombox, then you can take the entire board to the bathroom sink. Using very hot water, run and flush it inside the controls and if you have access to a tiny paint brush, you can insert it into the control and clean what you can. I have an oral irrigator which works great on flushing out the foreign debris inside the controls. Once the controls has completely dried off, you can use deoxit followed up with faderlube.
The best way to clean these controls is to disassemble them. Sometimes I get a rebuild request and recieve the board dripping wet with solvent that I presume to be deoxit. I take it to the sink and clean as much as I can first since I hate working on dirty board dripping with chemicals but invariably, the innards remain very dirty once disassembled. There simply is no way to do a thorough cleaning without disassemly. The chemicals, once introduced, also is very hard to completely remove. In general, these Alps sliders were never the best quality to begin with and I would not be suprised if one or more has failed internally. If they still work, disassembling them can sometimes cause the feelers to fall off so you should know this risk. Most that I've seen are pretty loose by now and barely holding on. Anyhow, once disassembled, I usually clean the resistor boards with hot water, soap, and toothbrush. The resistors do not build up oxidation. Oxidation will present on the feeler surfaces, not the carbon. The feelers are then thoroughly cleaned, and if loose, i will add some plastic specific superglue. Use only the ones with activator. After that, I mechanically strip the oxidation. Depending on how badly oxidized, this may involve polish compound, steel wool, or emery cloth. Use common sense and only drag it in the opposite direction the feelers are facing. After that, clean again to make sure no residue remains. At that point, I lube the slides only with silicone grease. You can use dielectric grease used on distributors. Old school grease monkeys will remember this compound in the days of points and condensors. Or you can find silicone grease at RC hobby stores, or some electronic stores. DO NOT use any on the carbon surfaces.
Before re-assembly, I brush the carbon tracks and feelers with Deoxit GL surface preservative. It feels like thick oil and helps to guard against future oxidation formation. DO NOT disassemble the controls unless you intend to do a complete rebuild. The reason is that the tabs holding the control together can only be bent so many times before they snap off. Once or twice is all they can take. If you're gonna stress those tabs, you might as well do it so it will last another 20 or 30 years as slider disassembly is not a maintenance process. Sometimes, people will ask me if I will rebuild their controls AFTER they've had a go at it already. Typically, I will say no. Not only are the tabs likely to all snap off but the pins also tend to break off if they weren't removed properly the first time around. I then replace all of the electrolytic capacitors. Rebuilt like this, I have found that the controls perform very well and noticeable sound improvement has frequently been observed. Also because there is no more internal residue, recurrance of intermittent operation is greatly reduced. Future applications of deoxit should involve no more than a drop or two, if ever needed again. Feel is usually restored to like-new.
Contrary to popular belief, the feel of the control has nothing to do with the feelers and carbon tracks. All the feel comes from the grease on the slides. People frequently flood the controls like the red sea thereby stripping off all the grease. Deoxit is a solvent. How else do you think it will remove oxidation? I've always warned to use deoxit sparingly but nobody listens. You only need to wet the surface between the contact and work it. More will not work better.
As for you other question regarding conductive grease..... WHY? You should not use any grease on the carbon tracks. Not only that but there are two ground strips that run along both of the carbon strips. The feeler bridges the ground to the respective location of the resistor. How do you think introducing a conductive material that covers everything is going to work out and affect the electrical signal path? Think about that for a moment.