I inadvertently discovered one factor some rubbers get sticky - oxidation
Our old house had a clean, mold-free basement that always seemed to smell damp to me, and all the regular steps you'd take to resolve it failed - a dehumidifier, fans/vents to bring in fresh air from outside, charcoal packets, copious amounts of DampRid, HEPA air filters - nothing worked.
So I bought a cheap ozone generator, and left it running for several full weekends we were away over the summer.
The good news was that it did actually get rid of the smell, which never came back afterwards.
But the bad news was that literally every soft rubber everything was suddenly sticky as hell (and all the cheaper, uncoated cheaper steel/aluminum things had a light but visible/slightly tacky tarnish, and any cheaper tapes were rendered into gooey, useless stripes).
Regular, 70% rubbing alcohol was helpful, but at the time we happened to have some 99% isopropyl alcohol, and that extra 29% made a surprising difference in effectiveness - like a night & day level difference.
Before we discovered that difference, I also tried a Mr Clean Magic Eraser - also surprisingly effective, and decidely easier to scrub with than an alcohol-soaked cloth.
Since then a friend who's knowledge of these things is superior to mine suggested talcum powder was also a highly effective way to remove that kind of stickiness, because it's both mildly abrasive and absorbent. IIRC he said to rub it in, leave it on then remove it a little later. I haven't actually tried this yet, but I trust his brain on matters like this.
I realize this doesn't answer your question directly, but on the off chance you may be using an ozone generator or negative ionizing air cleaner in a place your H4N Pro spends a lot of time, that could be a factor in accelerating the chemical leeching that causes it's recurrent stickiness.
