The $p$-exponent of the $K(1)_*$-local spectrum $\Phi SU(n)$

Let $p$ be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the $p$-exponent of the spectrum $\Phi SU(n)$is $(n-1) + \nu_p((n-1)!)$ for $n \geq 2$. It follows from this result that the $p$-exponent of $\Omega^{q} SU(n) \langle i \rangle$ is at least $(n-1) + \nu_p((n-1)!)$ for $n \geq 2$ and $i,q \geq 0$, where $SU(n) \langle i \rangle$ denotes the $i$-connected cover of $SU(n)$.