Cohomological collapse conjecture for finite cyclic semidirect products

Let $G$ be a finite cyclic group and let $L$ be a finitely generated $\mathbb ZG$-lattice. For the semidirect product extension

$$1\longrightarrow L\longrightarrow G\rtimes L\longrightarrow G\longrightarrow 1,$$

consider its Lyndon–Hochschild–Serre spectral sequence. Cohomological collapse conjecture. For every $k\geq 0$,

$$H^k(G\rtimes L,\mathbb Z)=\bigoplus_{i+j=k}H^i\bigl(G,H^j(L,\mathbb Z)\bigr).$$

This equality is the asserted consequence of collapse at $E_2$ for all finite cyclic groups and finitely generated integral lattices; the source reports collapse in all examples considered but does not establish the general statement.