Montesinos's four-dimensional branched covering conjecture

Let $M^4$ be a closed, oriented, connected $4$-manifold. A simple branched covering is a branched covering whose local monodromies around the branch set are transpositions; let $f:M^4\to S^4$ be a $4$-fold simple branched covering, with branch set $L\subset S^4$. Montesinos's conjecture. For every closed, oriented, connected $4$-manifold $M^4$, there exists a $4$-fold simple branched covering

$$f:M^4\to S^4$$

such that $L$ is an embedded surface in $S^4$. This conjecture extends the corresponding branched-covering representation results in dimensions three and four; the supplied text gives no evidence of a resolution, so its status remains open.