General weighted-homogeneous polynomial ideal conjecture

Let $k$ be an algebraically closed field, and give $x,y,z$ weights $1,2,3$, respectively. A polynomial is weighted-homogeneous if all of its monomials have the same weight. For an integer $m\geq 2$, let $g$ be a general weighted-homogeneous polynomial of weight $3m$, and set

$$I=(x,y)^{3m-2}+(g)\subset k[x,y,z].$$

General weighted-homogeneous polynomial ideal conjecture. The ideal $I$ contains no polynomial of standard degree $3m-3$. If true, the construction in the paper would imply the relevant primitive-line existence conjecture for $d=3m-1$ and sharpness of the genus bound in the corresponding degrees. The source presents this as the remaining consequence needed after its main theorem; no proof is supplied.