The composite-knot factor crossing-number conjecture

Let $K$ be a composite knot with factors $P_1,\ldots,P_r$, and let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. Composite-knot factor crossing-number conjecture. The crossing number of a composite knot is bigger than that of each of its factors; in the weaker stated variant, it is not less:

$$\operatorname{cr}(K)\geq \operatorname{cr}(P_i)\qquad(1\leq i\leq r).$$

The claim is presented as the intersection of the crossing-number additivity and satellite-knot crossing-number conjectures. It is not resolved in the paper.