Rank inequality for satellite knot Floer homology

Let $P$ be a pattern knot in $S^1\times D^2$, let $K$ be a companion knot in $S^3$, and let $U$ denote the unknot in $S^3$. Satellite rank inequality. There is an inequality

$$\dim \widehat{HFK}(P(K))\geqslant \dim \widehat{HFK}(P(U)).$$

This is the version for satellite knots of rank inequalities associated with peripheral-structure-preserving epimorphisms between knot groups. Its status is not established in the supplied source context.