  We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the
Hermite operator $\mathcal H=-\Delta+|x|^2$ on $\mathbb R^d$. Let $\Pi_\lambda$
denote the projection operator to the vector space spanned by the
eigenfunctions of $\mathcal H$ with eigenvalue $\lambda$. The optimal
$L^2$--$L^p$ bounds on $\Pi_\lambda$, $2\le p\le \infty$, have been known by
the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$,
we prove the optimal bound for the missing endpoint case. Our result is built
on a new phenomenon: improvement of the bound due to asymmetric localization
near the sphere $\sqrt\lambda \mathbb S^{d-1}$.
