Minimality conjecture for three-variable GT-systems

Let $d\ge 3$ be an integer and let $1\le n<m\le d-1$ satisfy $\operatorname{gcd}(n,m,d)=1$. Let $I^d_{0,n,m}\subset k[x,y,z]$ be the associated GT-system. Minimality conjecture. The GT-system $I^d_{0,n,m}$ is minimal. This conjecture is motivated by the preceding theorem, related results, and computations with Macaulay2; its general validity is not established here.