The defect-zero characterization of strongly quasipositive braids

Let $\mathcal{K}$ be a link type, let $\delta_3(\mathcal{K})$ be its defect, let $SL(\mathcal{K})$ be its maximal self-linking number, let $\chi(\mathcal{K})$ be its maximal Euler characteristic, and let $\beta(\mathcal{K})$ be its braid index. A braid representative $K$ of $\mathcal{K}$ realizes the braid index when its braid index is $\beta(\mathcal{K})$; it is strongly quasipositive in the usual sense.

Defect-zero characterization conjecture.

$$\delta_3(\mathcal{K})=0\quad\text{(equivalently }SL(\mathcal{K})=-\chi(\mathcal{K})\text{)}$$

if and only if there exists a braid representative $K$ of $\mathcal{K}$ such that $K$ realizes the braid index $\beta(\mathcal{K})$ and $K$ is strongly quasipositive.

This is identified in the source as the “Stronger Form of Conjecture 2” from the cited work. The paper provides the formulation but no resolution is supplied here.