Anti-pencil conjecture for the maximum number of non-negative-sum subsets

Let $A$ be a set of $n$ real numbers with total sum zero, and let $\mu_k(A)$ denote the number of $k$-subsets of $A$ with non-negative sum. Let $\mu_{\max}(k,n)$ be the maximum of $\mu_k(A)$ over all such sets $A$.

Anti-pencil conjecture. If $n\geq 4k$, then

$$\mu_{\max}(k,n)=\binom{n-1}{k}.$$

The lower bound is attained by choosing exactly one negative element, yielding a $k$-anti-pencil. Determining $\mu_{\max}(k,n)$ is described as a wide open problem, and the conjecture asserts optimality of this construction in the stated range.