Is the following consistent? $2^\omega > \omega_2$, and there is a normal precipitous ideal $I$ on $[\omega_2]^\omega$ such that every $X \subset [\omega_2]^\omega$ of size $< 2^\omega$ is in $I$?

Note:

(1) It is a theorem of Baumgartner-Taylor that every club subset of $[\lambda]^\omega$ has maximal size, $\lambda^\omega$, if $\lambda > \omega_1$.

(2) It is a theorem of Gitik that if $V \subset W$ are models of set theory in which $\kappa$ is regular in both and $\geq \omega_2$, and $r \in W \setminus V$ is a real, then $([\kappa]^\omega)^W \setminus ([\kappa]^\omega)^V$ is stationary in $W$.

(3) It is easy to get non-examples by adding reals with ccc forcing to a model with a precipitous ideal on $[\omega_2]^\omega$.

(4) One can show that the smallest normal ideal on $[\lambda]^{<\kappa}$, which makes every non-maximal-cardinality set have measure zero, is a proper ideal.

If this can be answered negatively for properties strengthening precipitousness, that would be interesting.