The satellite-knot crossing-number conjecture

Let $K$ be a satellite knot and let $P$ be its companion. Let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. Satellite-knot crossing-number conjecture. The crossing number of a satellite knot is bigger than that of its companion; in the weaker stated variant, it is not less:

$$\operatorname{cr}(K)\geq \operatorname{cr}(P).$$

This is a longstanding conjecture attributed in the cited literature to de Souza in its problem formulation. The paper notes that it contradicts the generic hyperbolicity conjecture, but gives no resolution of the conjecture itself.