Converse to the alpha-one theorem for Q-key polynomials

Let $\alpha$ be a symmetric weak composition, and let a P-key polynomial mean a polynomial of the form $\kappa^\mathsf{P}_\eta$ for a skew-symmetric weak composition $\eta$. Converse conjecture. If $\kappa^\mathsf{Q}_\alpha$ is a scalar multiple of a P-key polynomial, then

$$\alpha_1=0.$$

This is presented as the converse to the proved identity relating Q-key and P-key polynomials when $\alpha_1=0$ and is supported by computations; no resolution is given.