  Let $p>3$ be a prime number. We compute the rings of invariants of the
elementary abelian $p$-group $(\mathbb Z/p\mathbb Z)^r$ for $3$-dimensional
generic representations. Furthermore we show that these rings of invariants are
complete intersections rings with embedding dimension $\lceil r/2\rceil +3$.
  This proves a conjecture of Campbell, Shank and Wehlau in [CSW], which they
proved for $r=3$, and later Pierron and Shank proved it for $r=4$.
