  Perelman's doubling theorem asserts that the metric space obtained by gluing
along their boundaries two copies of an Alexandrov space with curvature $\geq
\kappa$ is an Alexandrov space with the same dimension and satisfying the same
curvature lower bound. We show that this result cannot be extended to metric
measure spaces satisfying synthetic Ricci curvature bounds in the
$\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane,
which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its
double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.
