Desargues configuration criterion for five-tuples of matrices

Let $A_1,\dots,A_5$ be real matrices whose degeneracy locus ${\mathcal X}:={\mathcal X}(A_1,\dots,A_5)$ is formed from the associated ten lines in ${\mathbb P}^3$. Desargues configuration conjecture. If ${\mathcal X}$ consists of ten real lines in ${\mathbb P}^3$ which intersect in ten points corresponding to a Desargues configuration, then $(A_1,\dots,A_5)$ are in the linear span of five fixed rank-one matrices. The conjecture gives a necessary condition for a degeneracy locus to satisfy the Desargues configuration; the supplied text does not indicate whether the condition has been proved or remains open.