Recursive structure conjecture for extremal polyominoes

Let $P$ be a $d$-dimensional polyomino with parameters $l_1,\dots,l_d$ and $v_1,\dots,v_d$, and suppose that the convex hull of $P$ has maximum volume. Let $P'$ be a subpolyomino of $P$. Recursive structure conjecture. The parameters satisfy $v_1=\dots=v_d=0$, and $P'$ can be chosen so that: (i) $P'$ has height $1$ in the direction of axis $i$; (ii) the projection of $P'$ along axis $i$ also has maximal convex-hull volume and parameters $l_1,\dots,l_{i-1},l_{i+1},\dots,l_d$; and (iii) $P$ can be decomposed into $P'$ and at most two orthogonal linear arms. This describes a recursive structure for maximizers, although the surrounding proof of the main theorem suggests that the assertion may already follow from established arguments in the paper.