Adem–Ge–Pan–Petrosyan collapse conjecture for semidirect products

Let $G\cong \mathbb{Z}/m$ be a finite cyclic group, let $L\cong\mathbb{Z}^n$ be a finitely generated free abelian group, let $\rho\colon G\to\operatorname{aut}_{\mathbb{Z}}(L)$ be a group homomorphism, and set $\Gamma=L\rtimes_{\rho}G$. The Lyndon–Hochschild–Serre spectral sequence associated to the semidirect product $L\rtimes_{\rho}G$ has terms arising from the extension $1\to L\to\Gamma\to G\to 1$. Adem–Ge–Pan–Petrosyan conjecture. The spectral sequence collapses in the strongest sense: all differentials in the $E_r$-term for $r\geq 2$ are trivial and all extension problems at the $E_{\infty}$-level are trivial. In particular, for all $k\geq 0$,

$$H^k(\Gamma;\mathbb{Z})\cong\bigoplus_{i+j=k}H^i(G;H^j(L)).$$

The paper proves the conjecture when the $G$-action on $L$ is free outside the origin, but gives an example with $n=6$ and $m=4$ in which the second differential does not vanish; thus the conjecture is false in general.