Key-positivity constraint for P-key expansions

Let $\alpha$ and $\gamma$ be weak compositions, with $\alpha$ skew-symmetric. Say that $\kappa^\mathsf{P}_\alpha-\kappa_\gamma$ is key positive when it is a nonnegative linear combination of key polynomials. Let $j$ be the first index such that $\gamma_j=0$. Key-positivity conjecture. If $\kappa^\mathsf{P}_\alpha-\kappa_\gamma$ is key positive, then

$$\gamma_i>1\qquad\text{for all }0<i<j.$$

The claim is suggested by calculations of the key polynomials occurring in P-key polynomials; the source gives no resolution.