Span-intersection conjecture for upper and lower roots

Let $W$ be a finite Coxeter group of rank $n$, let $c$ be a Coxeter element, and let $(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_{n-k})$ be a $c$-cluster, where the $\alpha_i$ are its lower roots and the $\beta_i$ are its upper roots. Let $\phi_c$ be the recursively defined map on roots, and write $\beta_i^{\perp}$ for the hyperplane orthogonal to $\beta_i$. Span-intersection conjecture. One has

$$\operatorname{Span}_{\mathbb{R}}(\phi_c^{-1}(\alpha_1),\ldots,\phi_c^{-1}(\alpha_k))=\beta_1^{\perp}\cap\cdots\cap\beta_{n-k}^{\perp}.$$

The two spaces have the same dimension, and the required orthogonality has been verified computationally for all Coxeter elements in Coxeter groups of rank at most $7$; proving the equality would remove the recursive map from the description of the associated noncrossing subspace.