Weak Hurwitz stability conjecture for consecutive type A minuscule polynomials

For each positive integer $n$, let $N_n(x)$ be the type A minuscule polynomial

$$N_n(x):=\frac{1}{4n+2}\sum_{k=1}^{n-1}k(n-k)\binom{2n+2}{2k+1}x^k.$$

A real polynomial is weakly Hurwitz stable if it is non-vanishing or identically zero when $\operatorname{Re}(x)>0$, equivalently, all its zeros lie in the closed left half-plane. Weak Hurwitz stability conjecture. The polynomial

$$N_{n+1}(x)^2-N_{n+2}(x)N_n(x)$$

is weakly Hurwitz stable for each positive integer $n$. The authors report computational verification for $n\leq 50$, but no general proof is given in the supplied text.