Melo–Winter's large-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

$$H^+(\infty,k)=\left(\bigcup_{n\geq k}H(n,k)\right)\setminus\{1,\dots,2^{k-1}\}.$$

Melo–Winter's large-intersection conjecture.

$$H^+(\infty,k)=\{2^{k-1}+2^i:i\in\{0,1,\dots,k-1\}\}.$$

The conjecture predicts that every possible intersection size larger than $2^{k-1}$ has one of these forms. The paper shows that this is almost true, but identifies an additional possible form, $35\cdot 2^{k-6}$, so the conjecture is refuted.