  We show that a uniform quadrangulation, its largest 2-connected block, and
its largest simple block jointly converge to the same Brownian map in
distribution for the Gromov-Hausdorff-Prokhorov topology. We start by deriving
a local limit theorem for the asymptotics of maximal block sizes, extending the
result in \cite{BFSS}. The resulting diameter bounds for pendant submaps of
random quadrangulations straightforwardly lead to Gromov-Hausdorff convergence.
To extend the convergence to the Gromov-Hausdorff-Prokhorov topology, we show
that exchangeable "uniformly asymptotically negligible" attachments of mass
simply yield, in the limit, a deterministic scaling of the mass measure.
