Hadamard matrices yielding orthogonal arrays satisfying the eight-column conditions

Let $m$ be an odd positive integer. A Hadamard matrix of order $4m$ is a square matrix with entries in $\{1,-1\}$ whose distinct rows are mutually orthogonal; an orthogonal array $\operatorname{OA}(4m,8,2,2)$ is an array with $4m$ rows and eight binary columns in which every pair of columns contains each binary ordered pair equally often. The conditions of Theorem are the conditions on such an array specified by that theorem.

The conjecture. For each odd $m$, there exists a Hadamard matrix $4m$ which yields an orthogonal array $\operatorname{OA}(4m,8,2,2)$ satisfying the conditions of Theorem.

This would provide the required eight-column constructions for every odd parameter and is stated to be stronger than the Hadamard conjecture. The supplied text gives computational and counting evidence motivating it, but does not report a resolution.