Ansensio–García-Marco–Knauer conjecture on induced degree-one subsets of the ternary Hamming graph

Let $H(n,3)$ be the Hamming graph on ${\mathbb Z}_3^n$, and let $U\subseteq {\mathbb Z}_3^n$ induce a subgraph of maximum degree $1$. Write $\alpha(H(n,3))$ for the independence number of $H(n,3)$.

Ansensio–García-Marco–Knauer conjecture. All such subsets satisfy

$$|U|=\alpha(H(n,3))+O(1).$$

The paper proves several bounds under additional hypotheses, including $|U|\leq \alpha(H(n,3))+729$ when $U$ is $i$-saturated for some $i\in[n]$. The conjecture asks whether a bounded additive error remains valid without any saturation assumption; this remains open.