Weak spatial mixing conjecture for BMS surveys

Consider the model on a regular tree of degree parameter $d$, with broadcast parameter $\theta$, and a BMS survey channel $W$. Write $P_e(W)$ for the survey error probability, and let WSM denote weak spatial mixing, namely convergence to zero of the expected supremum of the difference between the root observables induced by arbitrary pairs of boundary conditions, for every bounded continuous function on $\mathbb{R}\cup\{\pm\infty\}$. Weak spatial mixing conjecture. If $d\theta>1$, then there exists $\epsilon=\epsilon(d,\theta)>0$ such that, for every BMS survey $W$ satisfying

$$P_e(W)>\frac12-\epsilon,$$

WSM does not hold. The theorem preceding this conjecture proves the analogous statement for binary symmetric-channel surveys, while numerical evidence is cited for the extension to arbitrary BMS surveys with sufficiently large $P_e(W)$. The parser supplies no evidence that the conjecture has been resolved.