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Let $m$ be a positive integer, $X$ a graph with vertex set $\Omega$, and${\rm WL}_m(X)$ the coloring of the Cartesian $m$-power $\Omega^m$, obtained bythe $m$-dimensional Weisfeiler-Leman algorithm. The ${\rm WL}$-dimension of thegraph $X$ is defined to be the smallest $m$ for which the coloring ${\rmWL}_m(X)$ determines $X$ up to isomorphism. It is known that the ${\rmWL}$-dimension of any planar graph is $2$ or $3$, but no planar graph of ${\rmWL}$-dimension $3$ is known. We prove that the ${\rm WL}$-dimension of apolyhedral (i.e., $3$-connected planar) graph $X$ is at most $2$ if the colorclasses of the coloring ${\rm WL}_2(X)$ are the orbits of the componentwiseaction of the group ${\rm Aut}(X)$ on $\Omega^2$.

On the Weisfeiler-Leman dimension of some polyhedral graphs