SNP conjecture for Lascoux's Omega polynomials

Let $\alpha\in\mathbb Z_{\geq0}^{\infty}$, and let $\Omega_\alpha$ be the inhomogeneous analogue of the key polynomial obtained by replacing the Demazure operators with $\tau_i(f)=\partial_i(x_i(1-x_{i+1})f)$. A polynomial has SNP when every lattice point of its Newton polytope is an exponent vector. Omega-polynomial SNP conjecture.

$$\Omega_\alpha\text{ has SNP}.$$

The source gives computational evidence but no general proof.