SNP conjecture for Lascoux atoms

Let $\mathcal L_\alpha$ be the Lascoux atom obtained by replacing the Demazure operator in the definition of the key polynomial with $\widehat\tau_i=\tau_i-\operatorname{id}$, where $\tau_i(f)=\partial_i(x_i(1-x_{i+1})f)$. A polynomial has SNP when every lattice point of its Newton polytope is an exponent vector. Lascoux-atom SNP conjecture.

$$\mathcal L_\alpha\text{ has SNP}.$$

The source presents this as an open conjecture and supplies no general resolution.