Simple-knot detection conjecture for knots in lens spaces

Let $Y=L(p,q)$ be a lens space and let $K\subset Y$ be a knot. Here $\widehat{HFK}(Y,K)$ denotes the hat version of knot Floer homology, and a simple knot means a $(1,1)$ knot $W(p',q',r',s')$ with $q'=0$.

Simple-knot detection conjecture. If

$$\widehat{HFK}(Y,K)\cong \mathbb{Z}^p,$$

then $K$ is a simple knot.

This conjecture is suggested by the fact that the corresponding simple knot has knot Floer homology isomorphic to $\mathbb{Z}^p$ and is related to Berge's conjecture on knots in $S^3$ admitting lens-space surgeries. Its resolution status is not specified in the source.