The alternating Turán-type inequality for normalized Euler numbers

Let $E_n$ denote the Euler numbers and define the normalized Euler numbers by

$$\widetilde{E}_n=\frac{E_n}{n!}.$$

Alternating Turán-type inequality. For all $n\geq 0$ and $j,k\geq 1$, one has

$$(-1)^{n-1}\left[\widetilde{E}_n\widetilde{E}_{n+j+k}-\widetilde{E}_{n+j}\widetilde{E}_{n+k}\right]>0.$$

This inequality would simultaneously imply the contrasting strict log-concavity and strict log-convexity phenomena for the even and odd subsequences discussed in the paper. The supplied text presents it as appearing to be true, but gives no resolution, so its status is open.