Pemantle's CNA+ truncation stochastic-domination conjecture

Let $\mu\in\mathfrak{P}_n$ be a probability measure on subsets of $[n]$, and for $0\leq k\leq n$ let $\mu_k$ be its conditional measure given that the sampled set has size $k$. Write $\nu\preccurlyeq\mu$ when $\mu$ stochastically dominates $\nu$. Pemantle's truncation conjecture. If $\mu$ is CNA+ and

$$\mu(\{S\in 2^{[n]}:|S|=k\})\mu(\{S\in 2^{[n]}:|S|=k+1\})>0,$$

then $\mu_k\preccurlyeq\mu_{k+1}$. The source attributes this to Pemantle and the supplied status evidence indicates that it was subsequently confirmed, so the conjecture is solved.