Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components

Let $p$ be an odd prime, $q=p^k$, and $r=p^s$ with $s\mid k$. We study $\mathbb{F}_r$-valued generalized Maiorana-McFarland functions $H(x,y,z)=P^{(z)}(x)+\operatorname{Tr}_s^k(yz)$ on $V_n\times\mathbb{F}_q\times\mathbb{F}_q$. The ingredients $P^{(z)}$ are vectorial dual-bent functions with weakly