Even cycle decomposition thresholds for dense multipartite graphs

Let $r\geq 2$ be an integer. An $r$-partite graph $Γ$ with vertex partition $V_1,\ldots,V_r$ is $2$-balanced if there exists a positive integer $n$ such that $n\leq |V_i|\leq 2n$ for every $i\in[r]$. For an integer $\ell\geq3$, let $C_\ell$ denote the cycle of length $\ell$, and define $\hatδ(Γ)=\mi