The variance conjecture on projections of the cube

We prove that the uniform probability measure μ on every (n−k)-dimensional projection of the n-dimensional unit cube verifies the variance conjecture with an absolute constant CVarμ|x|2≤Csupθ∈Sn−1⁡Eμ〈x,θ〉2Eμ|x|2, provided that 1≤k≤n. We also prove that if 1≤k≤n23(log⁡n)−13, the conjecture is true for the family of uniform probabilities on its projections on random (n−k)-dimensional subspaces.