The free-subgroup conjecture for normal subgroups of Leavitt path algebra units

Let $E$ be a graph and $K$ a non-absolute field. Write $L_K(E)$ for the Leavitt path algebra of $E$ over $K$, and $L_K(E)^\times$ for its group of units. A subgroup is non-central if it is not contained in the center of $L_K(E)^\times$.

Free-subgroup conjecture. Every non-central normal subgroup of $L_K(E)^\times$ contains a non-cyclic free subgroup.

This conjecture extends the preceding results on solvable normal subgroups and non-central idempotents, predicting that every non-central normal subgroup has substantial nonabelian free-group structure. Its status is not specified in the source.