Update (and success story!).
I finally got the replacement contact pads from PacParts (I ordered a few different kinds as I wasn't sure what I needed; it wasn't in stock, but I got it after one month; I ended up only needing the long double-padded stripes, which apparently measure velocity = loudness of key strokes: http://www.pacparts.com/part.cfm?sku=91087274212 )
Here is the disassembly of my Casio PX-135 piano. The most time consuming was removing all keys, one by one, using two small flat screw drivers, and (as far as I recall) removing white keys first. This is after removing all keys:
Closer view of the problematic keys (marked with red). As you can see, no obvious issues here:
Next I removed the top layer of boards with single-padded contacts. (My test showed that these contacts are only responsible for the sustain feature, and have nothing to do with triggering a sound or its volume, so I knew they are not the culprits.)
Close-up on a bad key: nothing suspicious here either:
Finally when I removed the second layer of boards, I discovered stripes of double-padded contact stripes. I instantly saw the culprit: the key which I knew just got "stuck" at the highest volume had the corresponding double-padded contacts literally stuck in a pressed position. (Unfortunately I was so excited that I found the culprit that I forgot to take a picture of it.) Otherwise nothing obviously wrong with it - contact pads were clean and not worn out, no sticky stuff inside the pad. My only explanation is that perhaps these stripes are defective (may be too thin rubber; or perhaps wrong composition which deteriorated quickly).
The PacPart part which seemed to be identical was this one (Casio 91087274212; only 2$ each 12-key stripe): http://www.pacparts.com/part.cfm?sku=91087274212
It was very easy to replace the defective stripes (I had two stripes with bad keys): they are not glued or attached in any fashion to anything, they just lay in there. I lifted the bad stripes (grey colored), and put there the good ones (green colored, from PacPart):
After that my issue with stuck keys was gone.
If I knew what was the culprit in advance, I would have ordered replacement stripes for all the keys, because there is a good chance the rest are also defective which will eventually show. I will probably order the missing replacement stripes for the future, and if the problem returns I will replace the rest of them.
Dust couldn't possibly be the culprit here: the placement of the contact stripes is such (under double boards, and most importantly - with the contact pads facing down) that it is extremely unlikely for the dust to get there. (I am sure this is by design.) So my prior attempts to fix the issue using canned air were obviously doomed.
It's been one month since I fixed the piano, it was heavily used by me and my three kids, and everything seems to be perfectly fine.