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 $C$ $${\text{Var}}_{\mu }\mathrm{\mid }x{\mathrm{\mid }}^2\le C\underset{\theta \in S^{n-1}}{\sup }{\mathbb{E}}_{\mu }⟨x,\theta {⟩}^2{\mathbb{E}}_{\mu }\mathrm{\mid }x{\mathrm{\mid }}^2,$$ provided that $1\le k\le \sqrt{n}$. We also prove that if $1\le k\le n^{\frac{2}{3}}(\log n{)}^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces.