Generalized Desargues configuration criterion for matrix degeneracy loci

Let $A_1,\dots,A_{m+1}$ be linear operators on ${\mathbb R}^m$, and let ${\mathcal X}$ be their degeneracy locus in ${\mathbb P}^{m-1}({\mathbb R})$. The locus has expected dimension $m-3$ and degree $\binom{m+1}{2}$. Generalized Desargues configuration conjecture. The degeneracy locus ${\mathcal X}$ consists of $\binom{m+1}{2}$ $(m-3)$-dimensional linear subspaces with generalized Desargues configuration in ${\mathbb P}^{m-1}({\mathbb R})$ if and only if $(A_1,\dots,A_{m+1})$ are in the linear span of $m+1$ fixed rank-one matrices. The preceding results establish the implication from membership in such a span to the generalized Desargues configuration, while the converse is stated as a conjecture here and its resolution is not indicated.