Is this is the older button-snap style HRM-RUN with the little dude and the replaceable strap?
If so it's dying. Mine did the same thing -- it started the same way, but it would eventually cut it out after 20-30 minutes into a workout. Then it did it routinely and every time. I tossed it eventually and bought a new one -- problem disappeared.
It's NOT the battery -- I replaced mine with a known good, brand new battery -- no change.
These things fail in this sort of way and when they do they screw up your data for a *long* time too, as that bad data is in Connect and there's no way to get rid of it that I know of.
These straps do benefit from some tender loving care. And that begins with following the care instructions. If you have been diligent with that then, as noted above, chances are the strap's conductivity has been compromised. You're in luck if yours is the one with the snap-on module as the bands are readily available and cheap. If not, then it's a costly replacement.
You might also want to try replacing the battery in the HRM, if you haven't already tried this. Sometimes when the battery is low you can start to see some unusual readings.
If you still have the other button-snap style one, and it's still working, use that for a bit and see if the problem disappears.
It's almost-certainly not the strap part -- when mine did it I replaced the strap part with a new Polar strap (which fits perfectly) without improvement, and also changed the battery (although it was not indicating it was low on the watch, and it DOES report a low battery -- I had previously gotten a warning six months or so prior and had changed it then.)
My suspicion is that the seals on the electronics to the strap portion, over time, let a small amount of salty moisture (e.g. sweat) through and then it's just a matter of time before something bad gets bridged, and whatever is most-susceptible to that causes this specific sort of fault.