Euler-zigzag stability threshold conjecture for Eulerian polynomials

Let $A_m(x)$ denote the classical Eulerian polynomial, let $E_n$ be the $n$-th Euler zigzag number, and let a polynomial be Hurwitz stable when all of its zeros have negative real parts.

Euler-zigzag stability threshold conjecture. For any $n\geq 3$, the polynomial

$$(x+1)A_{n-1}(x)+kxA_{n-2}(x)$$

is Hurwitz stable if and only if

$$k>-2\frac{E_n}{E_{n-1}}.$$

The claim is motivated by computer evidence and is presented as an open question about the exact stability range of this one-parameter family.