Squared-binomial triangle satisfies conditions (C1) and (C2)

Consider the triangular array with entries $\binom{n}{k}^2$ for $0\leq k\leq n$. Conditions (C1) and (C2) are the two conditions imposed on a triangular array in the source's theorem on q-log-convexity. Squared-binomial triangle conjecture. The triangle

$$\left\{\binom{n}{k}^2\right\}$$

satisfies conditions (C1) and (C2) in that theorem. The claim is proposed in connection with the central binomial coefficients and central Delannoy numbers, but the supplied context does not state the conditions explicitly or report any verification.