Simple knot Floer homology implies grid number one

Let $L(p,q)$ be a lens space and let $K\subset L(p,q)$ be a knot. Write $\widehat{HFK}(L(p,q),K)$ for its hat knot Floer homology, and let $p$ be the order of $H_1(L(p,q))$.

Simple knot Floer homology conjecture. If

$$\operatorname{rk}\bigl(\widehat{HFK}(L(p,q),K)\bigr)=p,$$

then $K$ has grid number $1$.

The source says that this implication would establish the Berge conjecture after combining it with a previously proved surgery-to-simple-Floer-homology result. Its resolution is not specified in the source.