Normal curvature of immersed tori of dimension at most $18$

We prove that if the $n$-dimensional torus $T^n$ is smoothly immersed in the unit ball $B^q\subset {\mathbb{R}}^q$ and $n\le 18$, then there exists a point at which the spherical average of $\mid II(u,u){\mid }^2$ is at least $3n\mathrm{/}(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scalar-curvature obstruction for the torus with a conformal-Laplacian argument.