Jespers–Sun's nilpotent decomposition conjecture for integral group rings

Let $G$ be a finite group. Say that $G$ has nilpotent decomposition (ND) if, for every nilpotent element $n\in\mathbb{Z}G$ and every central idempotent $e\in\mathbb{Q}G$, one has $ne\in\mathbb{Z}G$. Write the Wedderburn–Artin decomposition as

$$\mathbb{Q}G=M_{n_1}(D_1)\oplus\cdots\oplus M_{n_\ell}(D_\ell),$$

where each $D_i$ is a division algebra; $G$ has at most one matrix component when at most one of the integers $n_i$ is greater than $1$.

Jespers–Sun's conjecture. A finite group $G$ has ND if and only if $\mathbb{Q}G$ has at most one matrix component.

The implication from having at most one matrix component to ND is immediate from the decomposition, since a division algebra has no nontrivial unipotent elements. The converse is the open direction, concerning whether ND can occur when multiple simple components have nontrivial matrix size.