The Apéry-sequence consecutive negativity conjecture

Let $A_n$ and $B_n$ be the Apéry numbers, let $e_n=2\cdot(\operatorname{lcm}\{1,2,\ldots,n\})^3$, and set

$$x_n=e_nA_n,\qquad y_n=e_nB_n.$$

Let $\Delta$ be the forward difference operator, $\Delta x_n=x_{n+1}-x_n$, and define

$$\delta_n=\Delta\left(\frac{\Delta y_n}{\Delta x_n}\right).$$

Apéry-sequence consecutive negativity conjecture. For every integer $N\geq2$, there is a positive integer $n$ such that

$$\delta_n,\delta_{n+1},\delta_{n+2},\ldots,\delta_{n+N}<0.$$

This is presented as a stronger numerical conjecture than the existence of an unbounded subsequence with negative $\delta_n$. The source gives no resolution.