The zero-partition maximization conjecture

For $\in\mathbb{N}$ and $\mathbf{x}\in\mathbb{N}^{n}$, a zero partition of $\mathbf{x}$ is a choice of signs $\sigma\in\{-1,1\}^{n}$ such that $\langle\mathbf{x},\sigma\rangle=0$. Zero-partition maximization conjecture. If $n$ is even, the number of zero partitions of $\mathbf{x}$ is at most the number for $(1,\ldots,1)$, namely

$$\binom{n}{n/2}.$$

If $n$ is odd, it is at most the number for a vector whose entries are all $1$ except one entry equal to $2$. The conjecture proposes extremal bounds for zero-sum sign assignments, but the supplied text gives no resolution or further context.