Reversal-Wilf conjecture for contraction containment in labeled trees

Let $\mathcal T_m$ denote the set of labeled trees on $[m]=\{1,\ldots,m\}$. For $T\in\mathcal T_n$ and $U\in\mathcal T_m$, write $T\preceq U$ if $U$ displays $T$, meaning that $T$ can be obtained from $U$ by contracting connected fibers and then standardizing the survivor labels. Define the support count by

$$\mu_T(m)=\#\{U\in\mathcal T_m:T\preceq U\}.$$

Let $R_n$ be label reversal, $i\mapsto n+1-i$. Reversal-Wilf conjecture. For $T_1,T_2\in\mathcal T_n$, if

$$\mu_{T_1}(m)=\mu_{T_2}(m)\qquad(m\ge n),$$

then $T_2=T_1$ or $T_2=R_n(T_1)$. Reversal of the ambient label set already gives $\mu_T(m)=\mu_{R_n(T)}(m)$ for all $m$, so the conjecture asserts that these are the only equal support-count sequences. It asks whether contraction containment among labeled trees has no Wilf-equivalences beyond label reversal.