First restore the natural balance
I am inclined to agree with Warpig's scepticism. Gunstocks that are very old (and I have a couple that are well past 200 years) have reached a natural equilibrium that is effectively permanent. When discussing refurbishment of old guns, it seems that too much attention is paid to the exterior surface - i.e. what you see - and not enough to the interior.
Now those who actually shoot their guns (warning - a bit of sarcasm coming!) may have noticed that the barrel gets not just warm, but if you shoot a competition series, pretty hot.
As a result, the barrel channel is not just dried out, but thoroughly dessicated. The dessication of the inside results in a stress between the moister outside and the drier inside that is probably a major source of the strain that warps a stock that was perfectly straight a century or so ago.
That is why those who have read my posts on refurbishment wil have observed that I advise that the barrel channel, receiver cutout etc. should be liberally soaked with linseed oil. On the outside, you can slobber on tru-oil, tung oil, BLO, whatever you like - it isn't going to penetrate to the inside in my lifetime. What is needed, before doing anything more drastic, is to restore the even moisture content throughout the wood from inside to outside.
The heat treatment shown in the films shots is a drastic way of achieving fast oil penetration in order to bend wood with a fairly thick cross-section. For a barrel channel, this is IMHO too harsh. Just try oiling the channel liberally first, and be patient.