Let $U\in V$ be an ultrafilter on $\omega$. We say $U$ is preserved under forcing with $\mathbb{P}$ if $\Vdash \forall x\subset \omega \ \exists Z\in U \ Z\subset x \vee Z\subset x^c$. In other words, $U$ generates an ultrafilter in $V[G]$. Familiar examples of such ultrafilters:

• Ramsey ultrafilters, P-points are preserved under Sacks forcing and Miller forcing (also their products)
• P-points/Ramsey ultrafilters are also preserved under iterations of Sacks forcing with countable support
• There are also combinatorial characterizations of ultrafilters preserved by Sacks forcing, related to Halpern-Lauchli theorem, see https://www.math.wisc.edu/~miller/res/ultra-s.pdf

My question is: is it known that an ultrafilter preserved by Sacks forcing necessarily needs to be preserved by side-by-side products of Sacks forcing (finite/countable support)? There are probably more ad-hoc examples of forcing $\mathbb{P}$ that preserves an ultrafilter $U$ but not the product $\mathbb{P}\times\mathbb{P}$ (is there any)? Maybe there is something easy that I overlook.