The finite-test-map factorization conjecture for topological spaces

Let $P$ be a finite set of maps between finite topological spaces. For a string of letters $l$ and $r$, let $(P)^{sl}$, $(P)^{slr}$, $(P)^{srl}$, and $(P)^{sr}$ denote the iterated orthogonality classes specified by that string, with $s$ indicating the relevant factorization construction. Finite-test-map factorization conjecture. Every map in the category of topological spaces decomposes as a map in $(P)^{sl}$ followed by a map in $(P)^{slr}$, and also as a map in $(P)^{srl}$ followed by a map in $(P)^{sr}$. The claim is presented as a proposed rule for a diagram-chasing calculus of formal topological spaces; the supplied text gives no resolution.