Recursive width conjecture for torus knots

Let $T_{p,q}$ denote the torus knot associated to positive coprime integers $p$ and $q$, and let $\widehat{HFK}(T_{p,q})$ be its knot Floer homology. Write $\operatorname{width}\widehat{HFK}(T_{p,q})$ for its width. Recursive width conjecture. If $p<q$, then

$$\operatorname{width}\widehat{HFK}(T_{p,q})-\operatorname{width}\widehat{HFK}(T_{p,q-p})=\left\lfloor\frac{(p-1)^2}{4}\right\rfloor.$$

Together with the base case $\operatorname{width}\widehat{HFK}(T_{1,q})=1$ and the symmetry $T_{p,q}=T_{q,p}$, this would recursively determine the width for every torus knot. The conjecture is known for $q\equiv\pm1\pmod p$ and for $p\leq6$, and computer computations verify it for $p,q<250$; its general validity remains open.