A distribution on triples with maximum entropy marginal

We construct an $S_3$-symmetric probability distribution on ${(a,b,c) \in $\mathbb{Z}$_{\geq 0}^3 : : : a+b+c =n }$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,\dots ,n\}$ with mean $n\mathrm{/}3$. Existence of such a distribution verifies a conjecture of Kleinberg, Sawin and Speyer, which is motivated by the study of sum-free sets.