Lascoux expansion conjecture for symplectic involution Grothendieck polynomials

Let $I^{\mathsf{fpf}}_\infty$ denote the fixed-point-free involutions, let $\mathfrak G^{\mathsf{Sp}}_z$ be the corresponding symplectic involution Grothendieck polynomial, and let $D^{\mathsf{Sp}}(\alpha)$ and $D^{\mathsf{Sp}}(z)$ denote the associated symplectic diagrams. Lascoux expansion conjecture. For every $z\in I^{\mathsf{fpf}}_\infty$, there is a set $\mathcal Z(z)$ of skew-symmetric weak compositions such that

$$\mathfrak G^{\mathsf{Sp}}_z=\sum_{\alpha\in\mathcal Z(z)}\beta^{|D^{\mathsf{Sp}}(\alpha)|-|D^{\mathsf{Sp}}(z)|}L^\mathsf{P}_\alpha,$$

so the coefficients belong to $\{\beta^n:n\in\mathbb N\}$. This generalizes the P-key expansion conjecture and was checked computationally through size $n\leq8$; no general resolution is given.