Minimal-diagram bridge-number realization conjecture for knots

Let $K$ be a knot. Let $\mathcal{C}(K)$ denote the set of crossing-number minimizing diagrams of $K$, and let $\mathcal{B}(K)$ denote the set of diagrams of $K$ realizing its bridge number.

Minimal-diagram bridge-number realization conjecture. For any knot $K$,

$$\mathcal{C}(K)\cap\mathcal{B}(K)\ne\emptyset.$$

The conjecture asserts that every knot has a crossing-number minimizing diagram whose bridge number is realized. It is proposed after examples showing that some minimal diagrams can have Wirtinger number strictly larger than the knot's bridge number, while an alternative minimal diagram realizes the bridge number.