Conjecture on nonsingular lifts of non-triangular definite matrices

Let $B$ be a non-triangular definite matrix in $T^\ell T^u$, where $T^\ell$ and $T^u$ are the lower- and upper-triangular submonoids generated by elementary matrices, and let $\operatorname{SLS}_n$ denote the relevant supertropical special linear monoid. Nonsingular-lift conjecture. If

$$B\in T^\ell T^u,$$

then

$$B\in\{A^\nabla: A\in \operatorname{SLS}_n\}.$$

The preceding discussion establishes the inclusion $\{A^\nabla:A\in\operatorname{SLS}_n\}\subseteq T^\ell T^u$ and notes that it is not known whether this inclusion is strict. The conjecture asserts the reverse inclusion for non-triangular definite matrices, thereby characterizing these matrices as quasi-inverses of matrices in the supertropical special linear monoid.