The conjectured max-domain-of-attraction limit for clustered Archimax copulas

Let $C_{\mathcal{G},\bm{\psi},\bm{\ell},Q}$ be a clustered Archimax copula satisfying the extended max-domain-of-attraction assumption and the standard stable tail dependence function assumption. For $k\in\mathcal{D}_1$, define $b_k=\operatorname{E}(1/Z_k^{\rho_k})$ with $Z_k\sim\operatorname{Beta}(1,d_k-1)$. Let $H$ be the multivariate limit law and $H_{ki}$ its marginal limit laws. The conjectured max-domain-of-attraction limit. One has $1/\bm{X}\in\mathcal{M}(H)$ and $1/X_{ki}\in\mathcal{M}(H_{ki})$, where $H_{ki}=\Phi_{\rho_k}$ for $k\in\mathcal{D}_1$ and $H_{ki}=\Phi_1$ for $k\in\mathcal{D}_2\cup\mathcal{D}_3$, and the stable tail dependence function of $H$ is, for all $\bm{x}\in\mathbb{R}_+^d$,

$$\ell_{\mathcal{G},\bm{\psi},\bm{\ell},Q}(\bm{x})=\operatorname{E}\left(\max_{\substack{k\in\mathcal{D}_1\\1\leq i\leq d_k}}\frac{x_{ki}W_k}{b_kS_{ki}^{\rho_k}}\right)+\sum_{k\in\mathcal{D}_2\cup\mathcal{D}_3}\ell_k(\bm{x}_k).$$

This is the proposed precise extension of the max-domain-of-attraction theorem to the boundary class $\mathcal{D}_3$; no proof or resolution is given in the supplied text.