Let ${f : E \rightarrow S}$ be a geometric morphism (between toposes).

For $s$ in $S$ and $x$ in $E$ let ${\pi : f^* s \times x \rightarrow x}$ be the obvious projection in $E$.

Let ${u \rightarrow f^* s \times x}$ be a complemented subobject of ${f^* s \times x}$.

Is the image of $u$ along $\pi$ complemented as a subobject of $x$?

(See also Images of complemented subobjects in hyperconnected toposes over Boolean bases)