No, this is not true. Let $G$ be the bowtie graph (this is the graph obtained by gluing two triangles at a vertex $u$). Then, $G$ does not have a spanning regular subgraph, but $\mathbb{1}$ is in the row space of $G$. Just set $x_e=\frac{1}{4}$ if $e$ is adjacent to $u$ and $x_e=\frac{3}{4}$ for the other two edges.
Tony Huynh
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