= Solution
The <universal cover> $\widetilde X$ is simply connected. Hence the closed combinatorial path $\gamma\sigma^{-1}$ is <null-homotopic> in its two-dimensional <CW complex>. The <van Kampen lemma> supplies a finite <disc diagram> mapping to $\widetilde X$ with precisely that boundary path.
Choose such a diagram with the smallest possible number of two-cells, and then with the smallest number of edges. If two adjacent cells formed a cancellable pair, removing that pair would preserve the boundary path and reduce the area, a contradiction. Thus
$$
\boxed{D_\sigma\to\widetilde X\text{ can be chosen reduced}.}
$$
This does not assume that $\widetilde Y$ is simply connected. Only the loop's null-homotopy in $\widetilde X$ is used. If the two boundary paths agree, a diagram with no two-cells is allowed; if they share initial or final segments, the diagram may have tree portions or cut vertices. A <disc diagram> need not be an embedded topological disk.
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