If all slices of a subset of a product have covering dimension $n$, does the set have dimension $2n$?

Question

Let $S\subset A\times B$ be closed, where $A$ and $B$ are closed balls of dim $n$.

Assume that $\forall b\in B$, $S\cap(A\times\{b\})$ has dimension $n$.

Then, does $S$ have dimension $2n$?

[Additional Question] If not, would there be any (not too strong) additional condition that makes this hold?