Zero-dimensionality conjecture for the systems $L_m$

For each even integer $m\geqslant24$, let $L_m$ be the algebraic set defined by the system of equations of order $m$.

Zero-dimensionality conjecture. $L_m$ is zero-dimensional for every even $m\geqslant24$.

If true for each fixed $m$, the paper's theorem would imply that there are at most finitely many prime powers $q$ with $q\equiv1\pmod m$ for which either $H_{q,m}$ or $M_{q,m}$ is a difference set in $\mathbb{F}_q$.