Local convergence conjecture for Schaeffer trees in large quadrangulations

Let $Q_n$ be a uniformly random rooted quadrangulation with $n$ faces, and let $z_n$ be a uniformly chosen vertex of $Q_n$. Let ${\rm Sch}_{z_n}(Q_n)$ denote the Schaeffer tree rooted at $z_n$, and let $(Q,{0\rm Sch}_\infty)$ denote the infinite random quadrangulation together with its infinite Schaeffer forest.

Local convergence conjecture. The pair

$$(Q_n,{0\rm Sch}_{z_n}(Q_n))$$

converges as $n\to\infty$ to

$$(Q,{0\rm Sch}_\infty)$$

in the sense of local weak convergence.

This conjecture formalizes the interpretation of the infinite forest as the local limit of a Schaeffer tree viewed from a uniformly chosen vertex in a large random quadrangulation. The supplied text gives no resolution, so it remains open.