The variance conjecture on projections of the cube

We prove that the uniform probability measure $\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\textrm{Var}_\mu|x|^2\leq C \sup_{\theta\in S^{n-1}}{\mathbb E}_\mu\langle x,\theta\rangle^2{\mathbb E}_\mu|x|^2, $$ provided that $1\leq k\leq\sqrt n$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces.