About an extension of the Davenport-Rado result to the Herzog-Schonheim conjecture for free groups

Let $G$ be a group and $H_1$,…,$H_s$ be subgroups of $G$ of indices $d_1$,…,$d_s$ respectively. In 1974, M. Herzog and J. Sch"onheim conjectured that if $\{H_i{\alpha }_i{\}}_{i=1}^{i=s}$, ${\alpha }_i\in G$, is a coset partition of $G$, then $d_1$,…,$d_s$ cannot be distinct. We consider the Herzog-Sch"onheim conjecture for free groups of finite rank and propose a new approach, based on an extension of the Davenport-Rado result for $G=\mathbb{Z}$.