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Consider almost disjoint families on regular $\kappa > \omega$ consisting only of stationary sets.

My question: Is there consistently an upper bound $<2^\kappa$ on the size of such a 'stationary' almost disjoint family (under suitable large cardinal assumptions)?

E.g. a Woodin cardinal implies the consistency of '$\text{NS}_{\aleph_1}$ is $\aleph_2$-saturated', which implies that any s.a.d. family has size $\leq \aleph_1$. However, as $X_i \cap X_j$ is not only non-stationary but bounded in $\kappa$, maybe weaker assumptions also imply the consistency of 'Every s.a.d. family on $\aleph_1$ has size $\leq \aleph_1$' ?

Consider almost disjoint families on regular $\kappa > \omega$ consisting only of stationary sets.

My question: Is there consistently an upper bound $<2^\kappa$ on the size of such a 'stationary' almost disjoint family (under suitable large cardinal assumptions)?

E.g. a Woodin cardinal implies the consistency of '$\text{NS}_{\aleph_1}$ is $\aleph_2$-saturated', which implies that any s.a.d. family has size $\leq \aleph_1$. However, as $X_i \cap X_j$ is not only non-stationary but bounded in $\kappa$, maybe weaker assumptions also imply the consistency of 'Every s.a.d. family on $\aleph_1$ has size $\leq \aleph_1$' ? (solved)

EDIT: The following cases for $\kappa$ remain open:

Always require $2^\kappa > \kappa^+$:

• $\kappa=\kappa^{<\kappa}$ and $\text{sad}< 2^\kappa$ ?
• $\text{sad} < \text{sat}(\text{NS}_\kappa)$ ?
• and, of course, $\text{sad} < \text{min} \{\text{sat}(\text{NS}_\kappa), \text{mad}\}$ ?

Consider almost disjoint families on regular $\kappa > \omega$ consisting only of stationary sets.

My question: Is there an upper bound $<2^\kappa$ on the size of such a 'stationary' almost disjoint family (under suitable large cardinal assumptions)?

E.g. a Woodin cardinal implies the consistency of '$\text{NS}_{\aleph_1}$ is $\aleph_2$-saturated', which implies that any s.a.d. family has size $\leq \aleph_1$. However, as $X_i \cap X_j$ is not only non-stationary but bounded in $\kappa$, maybe weaker assumptions also imply the consistency of 'Every s.a.d. family on $\aleph_1$ has size $\leq \aleph_1$' ?

Consider almost disjoint families on regular $\kappa > \omega$ consisting only of stationary sets.

My question: Is there consistently an upper bound $<2^\kappa$ on the size of such a 'stationary' almost disjoint family (under suitable large cardinal assumptions)?

E.g. a Woodin cardinal implies the consistency of '$\text{NS}_{\aleph_1}$ is $\aleph_2$-saturated', which implies that any s.a.d. family has size $\leq \aleph_1$. However, as $X_i \cap X_j$ is not only non-stationary but bounded in $\kappa$, maybe weaker assumptions also imply the consistency of 'Every s.a.d. family on $\aleph_1$ has size $\leq \aleph_1$' ?