Parabolic characterization conjecture for transitive quasiparabolic sets

Let $(X,\mathrm{ht})$ be a quasiparabolic $W$-set that is transitive and bounded below. For $x\in X$, let $\mathcal R_{\mathrm{ht}}(x)$ denote the set associated with the height function in the source. Let $(W^J,\ell)$ be the standard parabolic quasiparabolic $W$-set for $J\subset S$.

Parabolic characterization conjecture. If

$$|\mathcal R_{\mathrm{ht}}(x)|=1$$

for every $x\in X$, then

$$(X,\mathrm{ht})\cong (W^J,\ell)$$

for some $J\subset S$.

The claim characterizes the bounded transitive quasiparabolic sets with the stated property as parabolic ones. The source gives no resolution.