The faithful linear representation conjecture for p-compact groups

Let $p$ be a prime and let $B\Gamma$ be the classifying space of a $p$-compact group. A faithful linear representation means a map from $B\Gamma$ to $BSU(n)$, for some $n$, whose homotopy fibre is $\mathbb{F}_p$-finite. Linear representation conjecture. Every $p$-compact group has a faithful linear representation; more precisely, if $B\Gamma$ is regular, then there is a map

$$B\Gamma \longrightarrow BSU(n)$$

for some $n$ whose homotopy fibre is $\mathbb{F}_p$-finite.

The conjecture is used to make the class of finitely generated modules independent of the chosen normalization. The source states that it had in fact been proved: at the prime $2$ this follows from the classification of $2$-compact groups together with a construction for $BDI(4)$, and at odd primes it follows from the classification of $p$-compact groups with verification of the exotic cases.