Volume · Superchallenge problem

Stacks of $1$-cm cubes stand on the $9$ squares of a $3$ by $3$ grid (a square may have no cubes). Seen from the front, the tallest stacks in the left, middle and right columns are $3$, $1$ and $2$ cubes tall. Seen from the side, the tallest stacks in the front, middle and back rows are $2$, $3$ and $1$ cubes tall. What is the greatest number of cubes there could be?