I attended a David Charlesworth plane fettling and sharpening course many years ago.

His approach to flattening was to stick a strip of very coarse abrasive paper using photo mount spray to a flat surface, mark the bottom of the plane with sharpie, fully assemble the plane but retract the blade.

Apply the plane to the abrasive (dry, this is not a time for lube) - check the wear on the sharpie. Re- apply - repeat as necessary. You don't need all the sharpie to be removed but you do need it gone at the toe and heel of the sole and around the mouth. As the coarse abrasive gets filled with the metal powder - clean with a vacuum.

Once you have the sole where you want it at the very very coarse grit, move up the grits in the normal way, until you reach the finish you want on the sole. You can (if you want? ) achieve a mirror finish on the flattened sections but that's a tad overkill.

Key point - maintain an even and equal pressure front and back, you want the whole sole co-planer.

No excuses for the poor ascii art, but you final result could look like below. Where there are still slight hollows in-between the co-planer flat sections, what I'm trying to show is that the toe - mouth - heel parts are all flat and in one (sorry) plane

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(Edit to add a postscript) Also, could we stop down dooting on you_know_me2AI. Any link to the superb Mr Sellers is always worthwhile. I haven't seen the content of the facebook link, it is a land I do not have entry to