Let $M = \begin{pmatrix} k&0 \\ k&0 \\ 0 &0\end{pmatrix}$. Then $\inf_{\|x\|_3 = 1}\|Mx\|_3 = 0$. Whereas for any two dimensional subspace $V$, $0 < \sup_{\|y\|_{3/2} = 1, y\in V}\|M^T y\|_{3/2} \propto k$. So the proposed statement is false.