The heavy-handed Twyst-off conjecture

In the impartial combinatorial game Twyst-off, a position is a $P$ position if the next player has no winning move. Let heavy-handed Twyst-off be the variant on three stacks in which equal numbers of tiles may be removed from consecutive stacks only when removing from the larger of the two pairs sharing the same stack; all other moves are unchanged.

Heavy-handed Twyst-off conjecture. Heavy-handed Twyst-off has the same $P$ positions as Twyst-off on three stacks.

The conjecture is motivated by computational data and is used to compare the three-stack game with a more restrictive move rule. The supplied text gives no resolution, so its status remains open.