Hertweck's blockwise isomorphism conjecture

Let $G$ and $H$ be finite groups such that their integral group rings are isomorphic as rings,

$${\mathbb Z}G\cong {\mathbb Z}H.$$

Let $\operatorname{PCI}({\mathbb Q}G)$ denote the set of primitive central idempotents of ${\mathbb Q}G$.

Hertweck's blockwise isomorphism conjecture. For every $e\in\operatorname{PCI}({\mathbb Q}G)$,

$$Ge\cong He.$$

The statement is a blockwise refinement of the information preserved by an integral group-ring isomorphism. The source says that Hertweck formulated it in his Habilitationsschrift, but gives no resolution status.