  In this short note we observe that the concavity of Perelman's
$\mathcal{W}$-functional over a neighborhood of a K\"ahler-Ricci soliton inside
the space of K\"ahler potentials is a direct consequence of author's solution
of the variational stability problem for K\"ahler-Ricci solitons.
Independently, we provide a rather simple proof of this fact based on some
elementary formulas obtained in our previous work.
