Facet-defining conjecture for rigid tropical rays

Let $X$ be the schön very affine variety in the setup, with rigid ray $\tau\in \operatorname{Trop}(X)\cap \mathbb{R}_{\geq 0}^p$. Let $v_\tau$ be its ray vector and let $k_\tau=\operatorname{ord}_{E_\tau}(K_{X^\Sigma/X})+1$, where $E_\tau$ is the divisor corresponding to $\tau$. The LCT-polytope is

$$\operatorname{LCT}(G)=\left\{(s_1,\dots,s_p)\in\mathbb{R}_{\geq 0}^p\mid \sum_{j=1}^p (v_{\tau_i})_j s_j\leq k_{\tau_i}\text{ for }i=1,\dots,q\right\}.$$

Facet-defining conjecture. For every rigid ray $\tau\in \operatorname{Trop}(X)\cap\mathbb{R}_{\geq 0}^p$,

$$\{(v_\tau)_1s_1+\dots+(v_\tau)_ps_p=k_\tau\}$$

is facet-defining, meaning that its intersection with $\operatorname{LCT}(G)$ has dimension $p-1$ and lies in the boundary of $\operatorname{LCT}(G)$. This conjecture would identify the log-canonical-threshold hyperplane associated with every rigid ray as a genuine facet of the LCT-polytope, providing a natural affine translate of the Bernstein--Sato slope $\tau^\perp$.