de Gier's stationary distribution conjecture for half-turn-invariant FPL link patterns

Let $A_{\textsc{HT}}(N;\pi)$ be the number of half-turn-invariant FPLs of size $N$ with half-turn-invariant link pattern $\pi$, and let $A_{\textsc{HT}}(N)$ be the total number of such FPLs. Consider the Markov chain on half-turn-invariant link patterns obtained by choosing uniformly one of the symmetrized cyclic Temperley–Lieb generators; its unique stationary distribution is denoted by $\mu_{\textsc{HT}}$. de Gier's conjecture. The stationary distribution for half-turn-invariant link patterns of size $N$ is

$$\mu_{\textsc{HT}}(\pi)=\frac{A_{\textsc{HT}}(N;\pi)}{A_{\textsc{HT}}(N)}.$$

This conjecture extends the stationary-distribution claim from ordinary FPLs to the half-turn-invariant setting. The source gives no resolution evidence.