Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper A Segal conjecture for $p$-completed classifying spaces, it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$ is $p$-complete, but I have not succeeded in proving it. I hope this remains true when, more generally, $S$ is a $p$-toral group. Any suggestion or idea?, please.

Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper A Segal conjecture for $p$-completed classifying spaces, it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$ is $p$-complete, but I have not succeeded in proving it. I hope this remains true when, more generally, $S$ is a $p$-toral group (replacing $\Sigma^{\infty} BS$ by $(\Sigma^{\infty} BS)^{\wedge}_p$, because $\Sigma^{\infty} BS$ is no longer $p$-complete). Any suggestion or idea?, please.