Eventual uniqueness of maximum independent sets in monomial graphs

For positive integers $n$ and nonnegative integers $d$, let $G_n(d)$ be the graph whose vertices are the monomials of degree $d$ in $n$ variables, with two monomials adjacent exactly when their least common multiple has degree $d+1$. Let $\mathbf{a}_n(d)$ denote the number of maximum independent sets of $G_n(d)$. Eventual uniqueness conjecture. For any $n \geq 1$, there exists some $N \geq 0$ such that $\mathbf{a}_n(d)=1$ whenever $d\equiv 0\pmod n$ and $d\geq N$.

The conjecture extends the paper's established uniqueness results in selected cases, particularly for $n=3$ with $d$ divisible by $3$ and for $n=4$ with $d$ even. It predicts eventual uniqueness along the multiples of the number of variables for every $n$.