= Solution
Take the graphs $G_\zeta=\{(\delta,S_\zeta(\delta)):\delta<\kappa\}$. Each has size $\kappa$, and any two meet in fewer than $\kappa$ points, because their <functions> are eventually strictly ordered. Fix a <bijection> $b:\kappa\times\kappa\to\kappa$ and let $A_\zeta=b[G_\zeta]\in[\kappa]^\kappa$.
Given $I\in[\kappa^+]^\kappa$, enumerate it as $\{\zeta_\eta:\eta<\kappa\}$ and trim the sets in this order:
$$
A_{\zeta_\eta}^*=A_{\zeta_\eta}\setminus\bigcup_{\nu<\eta}A_{\zeta_\nu}.
$$
The deleted part is a union of fewer than $\kappa$ pairwise intersections, each of size less than $\kappa$. Regularity ensures its size is less than $\kappa$. If $\nu<\eta$, then $A_{\zeta_\eta}^*$ is disjoint even from the full earlier $A_{\zeta_\nu}$, hence from its trimmed version. Thus all the retained sets are pairwise disjoint and each lost fewer than $\kappa$ elements. This proves
$$
\boxed{*(\kappa,\kappa).}
$$
The argument is the <essential disjointness of small subfamilies> of a regular-cardinal almost disjoint family.
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