Disjoint refinements in $P(\kappa)/J$

Question

The following is a theorem of Baumgartner, Hajnal, and Mate:

Suppose $J$ is a normal ideal on $\omega_1$ which is nowhere $\omega_1$-dense. Then for any sequence $\langle A_\alpha : \alpha < \omega_1 \rangle$ of $J$-positive sets, there is a sequence $\langle B_\alpha : \alpha < \omega_1 \rangle$ of pairwise disjoint positive sets such that $B_\alpha \subseteq A_\alpha$ for all $\alpha$.

The proof is specific to $\omega_1$. But I conjecture that ZFC+CH proves the same statement with $\omega_1$ replaced by $\omega_2$ throughout.

Is the conjecture true? Are there any related results you think might shed light on this?

Answer 1

In my thesis I proved that my conjecture is false. It's not so simple-- the proof ties together chapter 2 and sections 5.1, 5.2, and 6.3.