McNamara–Sagan's infinite log-concavity conjecture for Pascal-triangle diagonals

Let $\mathcal{L}$ act on a sequence $A=\{a_0,a_1,a_2,\ldots\}$ by

$$\mathcal{L}(A)=\{a_0^2,a_1^2-a_0a_2,a_2^2-a_1a_3,\ldots\}.$$

A sequence is infinitely log-concave if every iterate $\mathcal{L}^j(A)$ is nonnegative. Let $u$ and $v$ be distinct nonnegative integers, and for each $n\geq 0$ consider the sequence $\{\binom{n+mu}{mv}\}_{m\geq 0}$.

McNamara–Sagan's conjecture. The sequence $\{\binom{n+mu}{mv}\}_{m\geq 0}$ is infinitely log-concave for all $n\geq 0$ if and only if $u<v$ or $v=0$.

This conjecture concerns diagonals of Pascal's triangle. The paper states that it was established by results of Brändén and Yu.