Conjecture on strong nodal domains of ternary Hamming graphs

Let $H(n,3)$ be the Hamming graph with vertex set $\mathbb{Z}_3^n$, where two vertices are adjacent when they differ in exactly one coordinate. Its Laplacian has eigenvalue $3n$ corresponding to the index $i=n$. For an eigenfunction $f$, let $\mathrm{SND}(f)$ denote its number of strong nodal domains.

Conjecture on ternary strong nodal domains. For any eigenfunction $f$ of $H(n,3)$, $n\geq 1$, with eigenvalue $3n$ we have $\mathrm{SND}(f)\geq n+1$.

This is the last remaining open case for the analogous problem with $q\geq 3$ in the paper. Numerical experiments for $n=2,3,4$ found minimum values $3,4,5$, respectively, supporting the conjectured linear lower bound.