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Let $W\subset GL(V)$ be a complex reflection group, and ${\mathscr A}(W)$ theset of the mirrors of the complex reflections in $W$. It is known that thecomplement $X({\mathscr A}(W))$ of the reflection arrangement ${\mathscr A}(W)$is a $K(\pi,1)$ space. For $Y$ an intersection of hyperplanes in $\mathscrA(W)$, let $X(\mathscr A(W)^Y)$ be the complement in $Y$ of the hyperplanes in$\mathscr A(W)$ not containing $Y$. We hope that $X(\mathscr A(W)^Y)$ is alwaysa $K(\pi,1)$. We prove it in case of the monomial groups $W = G(r,p,\ell)$.Using known results, we then show that there remain only three irreduciblecomplex reflection groups, leading to just eight such induced arrangements forwhich this $K(\pi,1)$ property remains to be proved.

On the $K(\pi, 1)$-problem for restrictions of complex reflection arrangements