Howroyd's periodicity conjecture for maximum independent sets

Let $G_3(d)$ be the graph on the monomials of degree $d$ in three variables, with two monomials adjacent exactly when their least common multiple has degree $d+1$. Let $\mathbf{a}_3(d)$ denote the number of maximum independent sets of $G_3(d)$. Howroyd's periodicity conjecture. The sequence $\{\mathbf{a}_3(d)\}_{d \geq 9}$ is $3$-periodic, with $\mathbf{a}_3(9)=1$, $\mathbf{a}_3(10)=27$, and $\mathbf{a}_3(11)=27$; equivalently, for $d \geq 9$,

$$\mathbf{a}_3(d)=\begin{cases}1,&d\equiv 0\pmod{3},\\27,&d\not\equiv 0\pmod{3}. \end{cases}$$

This conjecture, attributed to A. Howroyd through OEIS sequence A297557, concerns the eventual periodicity and exact values of the number of maximum independent sets. The paper presents it as motivation for studying these graphs; no resolution is supplied in the given text.