1-Rayleigh conjecture for the normalized Kahn–Saks polynomial

Let $x_1\leq\ldots\leq x_k$ be a chain, let $P_{\mathrm{KS}}$ be its Kahn–Saks polynomial, and let $N(P_{\mathrm{KS}})$ denote its normalization. For exponent vectors $\alpha$ and indices $i,j$, write $\partial^\alpha$ for the corresponding partial derivative. 1-Rayleigh conjecture. The normalized Kahn–Saks polynomial is $1$-Rayleigh: for every $(u,v)\in\mathbb{R}_{\geq0}^k$,

$$\partial^\alpha N(P_{\mathrm{KS}})\,\partial^{\alpha+e_i+e_j}N(P_{\mathrm{KS}})(u,v)\leq\partial^{\alpha+e_i}N(P_{\mathrm{KS}})\,\partial^{\alpha+e_j}N(P_{\mathrm{KS}})(u,v).$$

The conjecture proposes a strong Rayleigh-type inequality for the normalized Kahn–Saks polynomial. The excerpt supplies no resolution, so its status remains open.