Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids

Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the ass