The square-partition conjecture for Downright and LCTR game trees

Let $n$ be a positive integer, and let a partition of size $n^2$ be a partition whose diagram contains $n^2$ cells. For such a partition, consider the game trees of Downright and LCTR played on it, and let the number of nodes in a game tree mean its total number of nodes. Square-partition conjecture. The number of nodes in the game tree of any game of Downright and LCTR on a partition of size $n^2$ is maximized by the square partition % \begin{tikzpicture}[scale=0.08]% \draw (0,0) -- (0,-3)--(1,-3)--(1,0) (0,0)--(3,0)--(3,-1)--(0,-1) (0,-2)--(3,-2)--(3,-3)--(0,-3) (2,0)--(2,-3)--(3,-3)--(3,0);% \end{tikzpicture}% _{n,n}. This conjecture proposes that the square shape gives the largest game-tree complexity among partitions of the same size; the preceding rectangular calculation motivates it, but no resolution is supplied here.