Homotopy exponents of mod2r Moore spaces

We prove that $2^{r+1}\cdot {\pi }_{\ast }\left(P^m\left(2^r\right)\right)=0$ provided $m\ge 4$ and $r\ge 6$. This is the best possible result. As well, for $2\le r\le 5$ we obtain upper bounds on the homotopy exponent of $P^m\left(2^r\right)$.