The modified-weight characterization of kappa-hyperelliptic semigroups

Let $\mathrm{S}$ be a numerical semigroup of genus $g$, let $\kappa$ be a positive integer, and let $W_{\mathrm{K}}$ denote its modified weight. A semigroup is $\kappa$-hyperelliptic in the sense used above.

Modified-weight characterization. Whenever $g \gg \kappa$, $\mathrm{S}$ is $\kappa$-hyperelliptic if and only if

$${g-2\kappa \choose 2}+2\kappa \leq W_{\mathrm{K}} \leq {g-2\kappa \choose 2}+2\kappa^2.$$

This conjecture proposes an analogue of Torres' characteristic weight inequality with the modified weight $W_{\mathrm{K}}$ replacing the ordinary weight $W_{\mathrm{S}}$. The preceding results establish the corresponding equality in the bielliptic case $\kappa=1$, while the general assertion is presented as a speculation and remains open.