Nonvanishing conjecture for the middle homology of torus-knot diagrams

Let $D_{p,p}$ and $D_{p,p+1}$ be the torus-knot diagrams used in the paper, and let $H^{i,j}(D)$ denote their Khovanov homology in homological degree $i$ and quantum degree $j$. Nonvanishing conjecture. The rank of

$$H^{2p-2,p}(D_{p,p})$$

and, equivalently, the rank of

$$H^{2p-2,p}(D_{p,p+1}),$$

is nonzero. The statement is presented as a conjecture in the source, and the supplied context proves the equivalence of the two groups but does not resolve their nonvanishing.