  The main purpose of this work is to provide a framework for proving that,
given a family of random maps known to converge in the Gromov--Hausdorff sense,
then some (suitable) conditional families of random maps converge to the same
limit. As a proof of concept, we show that quadrangulations with a simple
boundary converge to the Brownian disk. More precisely, we fix a sequence
$(p_n)$ of even positive integers with $p_n\sim 2\alpha \sqrt{2n}$ for some
$\alpha\in(0,\infty)$. Then, for the Gromov--Hausdorff topology, a
quadrangulation with a simple boundary uniformly sampled among those with $n$
inner faces and boundary length $p_n$ weakly converges, in the usual scaling
$n^{-1/4}$, toward the Brownian disk of perimeter $3\alpha$.
