The cluster-count criterion for secant and initial ideals

Let $\mathcal{T}$ be a binary phylogenetic $[n]$-tree. For each $k$, let $c_k$ denote the number of $k$-clusters in $\mathcal{T}$, and let $J_{\mathcal{T}}$, $I_{2,n}$, $\omega$, and $\tau$ be as above, with

$$J_{\mathcal{T}}=\tau\cdot \operatorname{in}_{\omega}(I_{2,n}).$$

The cluster-count criterion. One has

$$\tau\cdot \operatorname{in}_{\omega}\left(I_{2,n}^{\{r\}}\right)=J_{\mathcal{T}}^{\{r\}}$$

if and only if

$$\sum_{k=2}^{r}(r-k+1)c_k<2r^2-2r.$$

This proposed modification is motivated by dimension bounds and examples showing failure of equality when the cluster-count expression is too large. The supplied text does not state whether the criterion has been proved or disproved.