The modified-weight characterization of kappa-hyperelliptic semigroups

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

Modified-weight characterization. Whenever $g \gg \ka$, $\sss$ is $\ka$-hyperelliptic if and only if

$${g-2\ka \choose 2}+2\ka \leq W_{\kk} \leq {g-2\ka \choose 2}+2\ka^2.$$

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