Atom-positivity conjecture for row division of key polynomials

Let $\alpha$ be a composition, and let $\kappa_\alpha$ denote the key polynomial indexed by $\alpha$. Let $\operatorname{\mathtt{rowDiv}}_k$ be the row operator. A polynomial is atom-positive if it is a non-negative integer linear combination of atom polynomials $A_\beta$.

Atom-positivity conjecture. The row operator applied to a key polynomial is atom-positive:

$$\operatorname{\mathtt{rowDiv}}_k(\kappa_\alpha)=\sum_\beta c_{\alpha,\beta}A_\beta,\qquad c_{\alpha,\beta}\in\mathbb{Z}_{\geq 0}.$$

The source gives an example whose key-polynomial expansion becomes atom-positive, providing evidence for the claim. No proof or resolution is supplied in the given text.