The polynomial inverse-factorization alternative for invertible noncommutative polynomial matrices

Let $\mathbb K\langle\underline{x}\rangle$ denote the free noncommutative polynomial algebra, let $\mathfrak r=(p_{ij})\in\mathbb K\langle\underline{x}\rangle^{L\times L}$ have noncommutative polynomial entries, and let $\underline{\mathfrak a}=(\mathfrak a_1,\ldots,\mathfrak a_d)\in\mathcal A^d$. Assume that

$$\mathfrak r(\underline{\mathfrak a})=(p_{ij}(\underline{\mathfrak a}))_{i,j}\in\mathcal A^{L\times L}$$

is invertible. Polynomial inverse-factorization conjecture. There exist either noncommutative rational expressions $S_{11},\ldots,S_{LL}$ satisfying

$$\underline{\mathfrak a}\in\bigcap_{i,j}\operatorname{dom}_{\mathcal A}(S_{ij}),$$

or noncommutative polynomials $S_{11},\ldots,S_{LL}$, such that $(S_{ij}(\underline{\mathfrak a}))_{i,j}$ is invertible and there exist $1\leq i,j\leq L$ for which

$$\sum_{k=1}^L p_{ik}(\underline{\mathfrak a})S_{kj}(\underline{\mathfrak a})$$

is invertible in $\mathcal A$. The assertion is presented as an alternative formulation needed to prove the preceding matrix-inverse conjecture.