Sullivant's secant–initial ideal conjecture for Plücker tree ideals

Let $d4b1$ be a binary phylogenetic $[n]$-tree, let $d714 d7d9 \in \mathbb{R}^{\binom{n}{2}}$ be a weight vector, and let $d70f \in \{\pm1\}^{\binom{n}{2}}$ be a sign vector such that

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

Sullivant's conjecture. For every $r$, one has

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

The conjecture asks whether taking secant ideals commutes with taking the corresponding signed initial ideal for Plücker tree ideals. It is attributed in the source to Sullivant's Conjecture 7.10; no resolution is supplied here.