The optimal functional $2^{s-1}$-batch code conjecture

A functional $k$-batch code of dimension $s$ consists of servers storing linear combinations of $s$ linearly independent information bits, with every multiset of $k$ linear combinations recoverable from disjoint subsets of servers. Let $FB(s,k)$ be the minimum number of servers in such a code.

Optimal functional $2^{s-1}$-batch code conjecture. For any $s$,

$$FB(s,2^{s-1})=2^s-1.$$

This asserts that the minimum number of servers for $k=2^{s-1}$ is exactly the number of nonzero vectors in $\mathbb{F}_2^s$. The source presents it as open; the subsequent pairing result is an approach toward this conjecture.