Melo–Winter's small-intersection conjecture for hypercubes

Let $\mathbb{H}^n=\{0,1\}^n$ be the vertices of the $n$-dimensional hypercube, and let $H(n,k)$ be the set of intersection sizes $|\mathbb{H}^n\cap S|$ as $S$ ranges over the $k$-dimensional linear subspaces of $\mathbb{R}^n$. Define

$$H(\infty,k)=\bigcup_{n\geq k}H(n,k),\qquad H^-(\infty,k)=H(\infty,k)\cap[2^{k-1}],$$

where $[r]=\{1,\dots,r\}$. Melo–Winter's small-intersection conjecture.

$$H^-(\infty,k)=[2^{k-1}].$$

The conjecture asserts that every positive integer at most $2^{k-1}$ occurs as an intersection size. The paper proves that this conjecture is false, showing that a positive fraction of the small values is missing.