Connectivity conjecture for the infinite Schaeffer forest

Let $Q$ be the infinite random quadrangulation, and let ${\rm Sch}_\infty(Q)$ be the spanning forest obtained by applying the Schaeffer construction to the increments of the Busemann function $h$ defined by

$$\Delta h(x,y)=\lim_{z\to\infty}d(x,z)-d(y,z)=h(x)-h(y).$$

Connectivity conjecture. The graph ${\rm Sch}_\infty(Q)$ is connected almost surely.

This conjecture interprets the forest seen from infinity as a single limiting Schaeffer tree. The supplied text gives no resolution, so its status remains open.