Zupan's distance conjecture for bridge positions of links

Let $K$ be a $c$-component link embedded in a compact $3$-manifold $M$ with $k$ $S^1\times S^2$ summands. For a $(g,b)$-bridge position of $(M,K)$, let $D(\Sigma)$ denote the distance between the associated disk sets.

Zupan's distance conjecture. For any $(g,b)$-bridge position of $(M,K)$,

$$D(\Sigma)\geq g-k+b-c.$$

Moreover, equality holds if and only if $K$ is an unlink in $\#^k S^1\times S^2$.

This conjecture would merge the stated lower-bound result with the exact calculation for unlinks. It extends the known distance estimate for knots in manifolds without $S^1\times S^2$ summands, while the claimed equality characterization remains open.