The following is due to Alan Dow:

In any model obtained by adding $\aleph_2$ many Cohen reals to a model
of $\mathsf{CH}$ the statement is false.
We force with $\mathbb{P}=\operatorname{Fn}(\omega_2,2)$ and we let
$\dot{\mathcal{I}}$, $\dot{\mathcal{J}}$ and $\dot u$ be 
$\mathbb{P}$-names such
that $\dot{\mathcal{I}}$ and  $\dot{\mathcal{J}}$ are forced to be ideals
and $\dot u$ is forced to be the unique ultrafilter that extends the
two associated regular filters.

Now let $M$ be an elementary substructure of a suitable large $H(\theta)$
that has cardinality $\aleph_1$ and that is closed under $\omega$-sequences.
Let $\delta=M\cap\omega_2$ and 
$\mathbb{P}_M=\operatorname{Fn}(\delta)$.

By elementarity the $\mathbb{P}_M$-names
$\dot{\mathcal{I}}\cap M$ and  $\dot{\mathcal{J}}\cap M$ are forced to be 
ideals and $\dot u\cap M$ is forced to be the unique ultrafilter that extends 
the two associated regular filters.

Work in $V[G\cap M]$.
We write $\mathcal{I}_M$, $\mathcal{I}_M$ and $u_M$ for the interpretations
of $\dot{\mathcal{I}}\cap M$, $\dot{\mathcal{J}}\cap M$ and $\dot u\cap M$
in this model.

Note that every element of $u_M$ meets elements of $\mathcal{I}_M$ 
and $\mathcal{J}_M$.
On the other hand if $\mathcal{I}'$ and $\mathcal{J}'$ are countable 
subfamilies of $\mathcal{I}_M$ and $\mathcal{J}_M$ respectively then some
infinite subset, $x$, of $\omega$ separates the elements of $\mathcal{I}'$ from
those of $\mathcal{J}'$.
This implies that, without loss of generality, for every countable subfamily
$\mathcal{J}'$ of $\mathcal{J}_M$  there is an element $x\in u_M$ such that
$x\cap y$ is finite for all $y\in\mathcal{J}'$.

Using this one can construct a sequence 
$\langle b_\alpha:\alpha<\omega_1\rangle$
in $\mathcal{J}_M$
such that every element of $u_M$ contains all but countably many of 
the $b_\alpha$. 

In  $V[G]$ consider the next Cohen real $c$, added by 
$\operatorname{Fn}([\delta,\delta+\omega,2)$, say and assume, 
without loss of generality, that $c\notin u$.

There must be a set $Y\subseteq c$ that separates 
$\lbrace x\cap c:x\in\mathcal{I}\rbrace $ from $\lbrace y\cap c:y\in\mathcal{J}\rbrace $.
In $V[G\cap M]$ we take names, $\dot c$ and $\dot Y$, for $c$ and $Y$,
note that these are $\operatorname{Fn}(C,2)$-names for some countable set $C$.
For every $\alpha<\omega_1$ it is forced that $\dot Y\cap b_\alpha$ is finite;
by pigeon-holing there will be one $p\in\operatorname{Fn}(C,2)$ and one 
$n\in\omega$ such that $p$ forces $\dot Y\cap b_\alpha\subseteq n$ for 
uncountably many $\alpha$.
Still in $V[G\cap M]$ let 
$Y_p=\lbrace k:(\exists q\le p)(q \Vdash k\in\dot Y)\rbrace $.
Then $b_\alpha\cap Y_p\subseteq n$ for uncountably many $\alpha$ so that
$Y_p\notin u_M$.
On the other hand for every $x\in\mathcal{I}_M$ it is forced that 
$x\cap\dot c\subseteq \dot Y$ (mod finite); this implies that 
$x\setminus Y_p$ is finite for all $x\in\mathcal{I}_M$, so that $Y_p\in u_M$.