Blockwise spectrum problem for integral group rings

Let $G$ be a finite group. Let $V({\mathbb Z}G)$ be the normalized unit group of ${\mathbb Z}G$, let $\operatorname{PCI}({\mathbb Q}G)$ be the set of primitive central idempotents of ${\mathbb Q}G$, and let $\operatorname{spec}(A)$ denote the spectrum of a finite group or its corresponding block. For $e\in\operatorname{PCI}({\mathbb Q}G)$, write $Ge$ and $V({\mathbb Z}G)e$ for their projections to the block determined by $e$.

Blockwise Spectrum Problem. For every primitive central idempotent $e\in\operatorname{PCI}({\mathbb Q}G)$,

$$\operatorname{spec}(Ge)=\operatorname{spec}(V({\mathbb Z}G)e).$$

This is a blockwise weakening of the spectrum problem: it asks for equality of the spectra after projection to each primitive central block, rather than only a global statement.