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

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. For a fixed integer $k\geq 0$, consider the sequence $\{\binom{n+k}{k}\}_{n\geq 0}$.

McNamara–Sagan's conjecture. The sequence

$$\left\{\binom{n+k}{k}\right\}_{n\geq 0}$$

is infinitely log-concave for every fixed $k\geq 0$.

The conjecture was made for columns of Pascal's triangle. It is trivial for $k=0,1$ and is solved affirmatively for $k\geq 2$ by the theorem proved in the paper.