Amicability–anti-amicability permutation conjecture for Clifford basis matrices

Let $m\geqslant 0$. Consider the $4^m$ canonical basis matrices of the real representation of the Clifford algebra $\mathbb{R}_{m,m}$; a pair has disjoint support when the supports of its two matrices are disjoint, and is amicable or anti-amicable according as the corresponding product is symmetric or skew. Amicability–anti-amicability permutation conjecture. There is a permutation $\pi$ of the set of $4^m$ canonical basis matrices that sends every amicable pair with disjoint support to an anti-amicable pair, and vice versa. The conjecture concerns a global symmetry exchanging the two edge types in the restricted amicability/anti-amicability graph; its resolution is not supplied here.