Forest sufficient condition for polynomial approximation homomorphism complexity

Let $F$ and $H$ be graphs. Let an $F$-forest $T$ be the forest construction used in the source, and let $T^{\star}$ denote its associated graph. Forest sufficient-condition conjecture. If there exists an $F$-forest $T$ such that

$$T\rightarrow F\qquad\text{and}\qquad H\rightarrow T^{\star},$$

then

$$M_{F,H}(\varepsilon)=\operatorname{poly}(1/\varepsilon).$$

This is presented immediately after the conjectural characterisation of polynomial versus exponential behaviour and is intended as the polynomial side of that dichotomy. The supplied text does not establish this statement as a theorem, and the conjecture remains open.