Leading-term conjecture for P-key polynomials

Let $\alpha$ be a skew-symmetric weak composition, let $n=\ell(\alpha)$, and let $\widetilde\rho(\alpha)$ and $\widetilde\gamma(\alpha)$ denote its strict sub-diagonal row and column counts. Leading-term conjecture. If $\widetilde\rho(\alpha)\neq\widetilde\gamma(\alpha)$, then

$$\kappa^\mathsf{P}_\alpha\in x^{\widetilde\rho(\alpha)}+x^{\widetilde\gamma(\alpha)}+\sum_{\widetilde\rho(\alpha)<_{\mathrm{lex}}\delta\in\mathbb{N}^n}\mathbb{N}x^\delta.$$

This is presented as the symplectic analogue of the preceding Q-key conjecture and is supported by computations; no resolution is given.