  We prove that the Parker--Semeraro systems satisfy six of the nine
Kessar--Linckelmann--Lynd--Semeraro weight conjectures for saturated fusion
systems. As a by-product we obtain that Robinson's ordinary weight conjecture
holds for the principal $3$-block of Aut$(G_2(3))$, the principal $5$-blocks of
$HN$, $BM$, Aut$(HN)$, $Ly$, the principal $7$-block of $M$, and the principal
$p$-blocks of $G_2(p)$ for $p\geq 3 $.
