Rings of invariants for three dimensional modular representations

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 $r=4$.