Nonvanishing conjecture for sesquicuspidal curves in the complex projective plane

Let $d$ be an integer and let $p/q>1$ be a reduced fraction. The count $\mathbf{T}_{d}^{p/q}$ is the index-zero finite-energy holomorphic-plane count associated with degree $d$ curves in $\CP^2$. Nonvanishing conjecture. For any reduced fraction $p/q>1$ satisfying

$$p+q=3d$$

and

$$(p-1)(q-1)\leq(d-1)(d-2),$$

we have $\mathbf{T}_{d}^{p/q}\neq0$. This is equivalent to the existence of an index-zero $(p,q)$-sesquicuspidal symplectic curve in $\CP^2$ of genus zero and degree $d$ whenever permitted by the adjunction formula. An affirmative answer would imply optimality of Hind's folding embedding in the range described in the source; the conjecture's resolution is not established here.