Strengthened varentropy conjecture for finite positive monotone concave sequences

Let $(y_n)_{n=1}^N$ be a finite positive monotone and concave sequence, meaning

$$y_n\geq\frac{y_{n-1}+y_{n+1}}{2},\qquad 1<n<N.$$

For every $\gamma>0$, define

$$K(t)=(t+\gamma)\sum_{n=1}^N y_n^{t/\gamma}.$$

Strengthened varentropy conjecture. The function $K$ is log-concave: $\log K(t)$ is concave on $(-\gamma,+\infty)$.

This is proposed as a strengthening of the preceding conjecture, motivated by the analogous continuous result. The source does not state that it has been proved or disproved.