Connectivity conjecture for tilted Bruhat posets

Let $\mathbf{a}\in[n]^n$. Consider the poset defined by $\ell_{\mathbf{a}}(w)$ and the relation $\lesssim_{\mathbf{a}}$, and let $\sim_{\mathbf{a}}$ denote the associated equivalence relation. A poset is connected when its Hasse diagram is connected.

Connectivity conjecture for tilted Bruhat posets. The poset defined in the source by $\ell_{\mathbf{a}}(w)$ is connected, and $\lesssim_{\mathbf{a}}$ is connected when restricted to each equivalence class under $\sim_{\mathbf{a}}$.

Connectivity would describe the global structure of the tilted orders and their decomposition into equivalence classes. The supplied source does not provide evidence of resolution.