The ULC conjecture for competing urn measures

Let $m$ balls be assigned independently to urns $1,\ldots,n$, and let $X_i$ indicate whether urn $i$ is occupied. The law of $(X_1,\ldots,X_n)$ is a competing urn measure. Its rank sequence is $(\mu(|\eta|=i))_{i=0}^n$, and ULC means that the binomially normalized rank sequence is log-concave with no internal zeros. Competing-urn conjecture. Every competing urn measure is ULC. The source notes that competing urn measures are known to be NA, but presents the stronger ULC assertion as conjectural.