Integer-root conjecture for total domination polynomials

Let $G=(V,E)$ be a simple graph, let

$$D_t(G;x)=\sum_{i=1}^{n} d_t(G,i)x^i$$

be its total domination polynomial, where $n$ is the order of $G$ and $d_t(G,i)$ counts the total dominating sets of cardinality $i$. A root of $D_t(G,x)$ is called a total domination root. The integer-root conjecture. If $r$ is an integer root of $D_t(G,x)$, then

$$r\in\{-3,-2,-1,0\}.$$

This proposes a restriction on the possible integer total domination roots. Earlier work established the narrower possibilities $-2$ and $0$ for domination-polynomial roots under a minimum-degree hypothesis, but the corresponding unrestricted claim for total domination polynomials remains open.