Weak same-level reducibility conjecture for involutions

Let $m\geq 2$ be even, let $R\in SO(p)$ be an involution, and suppose that $R$ has level $m$. A sign-change matrix is a matrix in $\mathcal I_p^+$, and $R$ is sign-change reducible if it is reducible by such a sign change. Weak same-level reducibility conjecture. If $R$ is sign-change reducible, then it is reducible by a sign-change matrix of level $m$.

The preceding discussion explains that this is weaker than requiring all minimizers of the distance to have the same level as $R$. It is intended to simplify the analysis of sign-change reducibility by restricting attention to sign changes of the same level; the source does not report a resolution.