The middle-regime doubling conjecture for distinct rectangle tilings

Let $T_0,T_1,T_2,\dots$ denote the threshold sequence used to define the middle regimes, and let $W^*(n,m)$ be the minimum internal wall measure for tilings of an $n\times n$ square by $m$ rectangles with pairwise distinct dimensions. The middle regime for $m$ is

$$T_{m-2}\le n<T_{m-1}.$$

Middle-regime doubling conjecture. For every $m\ge4$ and every $n$ in this middle regime,

$$W^*(n,m)=n+2(m-2).$$

The formula is established in the source for $m\in\{4,5\}$ by case analysis and for $m=6$ by exhaustive certification; the conjecture remains open for general $m\ge7$.