The matrix inverse realization conjecture for noncommutative rational functions

Let $\mathcal A$ be the algebra under consideration and let $L\geq 1$. For an invertible matrix $\mathfrak A\in\mathcal A^{L\times L}$, write

$$\mathfrak A=(\mathfrak a_{ij})_{1\leq i,j\leq L},\qquad \mathfrak A^{-1}=(\mathfrak b_{ij})_{1\leq i,j\leq L}.$$

Matrix inverse realization conjecture. Each entry of $\mathfrak A^{-1}$ is a noncommutative rational expression in the entries of $\mathfrak A$: there exist noncommutative rational expressions $S_{ij}$, for $1\leq i,j\leq L$, such that

$$(\mathfrak a_{11},\ldots,\mathfrak a_{LL})\in\operatorname{dom}_{\mathcal A}(S_{ij})$$

and

$$\mathfrak b_{ij}=S_{ij}(\mathfrak a_{11},\ldots,\mathfrak a_{LL}),\qquad 1\leq i,j\leq L.$$

This assertion is stated as equivalent to the preceding conjecture concerning inversion of matrix-valued noncommutative rational functions; its resolution would provide rational-expression realizations of inverse matrix entries over the algebra.