Uniform diameter bound for chess squares of the Feigenbaum map

Let ${\cal P}_0$ be the collection of chess squares in the level-zero puzzle for the Feigenbaum map, and let $\operatorname{diam}(P)$ denote the diameter of a chess square $P$. Uniform diameter-bound conjecture. There exists a constant $K\in\mathbb{R}$ such that every chess square $P\in{\cal P}_0$ satisfies

$$\operatorname{diam}(P)\leq K.$$

This is presented as a stronger result than the preceding conjecture on the uniform separation of nested chess squares; the source gives no proof or resolution, so the problem remains open.