[morzh]

Which mod are we talking about? Oh, the extra RC filter before the R3?

We could, but before we start modding things...

1. We do know that the radio probably did not leave the factory with this problem. So introducing the mods before understanding the reason behind the problem has less chances of working, and even if it does, we are masking the problem, by fixing what we still have not understood.

2. I would suggest this. Put a scope to the R2 (the one across the coil), to the pin that is the opposite to the one that connects to the R3.
Tune into the station.
Then connect the scope. Make sure it does not ruin the reception. It should be your "good behavior" reception.
Make sure you could see the oscillation well, sync the scope on it to make it stable. Note (write down/photograph) the frequency, amplitude, jitter (if possible, otherwise just the stability of it).

Then wait 'till the crap-out.
Note, again, the signal's parameters: the amplitude, frequency, jitter.

If the scope is digital, it is easy; else you will have to calc it yourself.

3. If no crap-out happens, it could be possible your scope GND is doing it. Then connect the scope through an isolation xfmr.

Then we will proceed, depending on the results.