Finite principal-specialization conjecture for chromatic symmetric functions of trees

Let $N$ be a positive integer, let $q$ be an indeterminate, and let $T$ and $U$ be trees with at most $N+1$ vertices. Write $X_T(1,q,q^2,\ldots,q^N)$ for the principal specialization of the chromatic symmetric function of $T$.

Finite principal-specialization conjecture. For all such trees,

$$T\cong U \quad\text{if and only if}\quad X_T(1,q,q^2,\ldots,q^N)=X_U(1,q,q^2,\ldots,q^N).$$

The parser provides no resolution evidence for this finite version; it is presented in the paper as a conjectural analogue arising from principal specialization.