NP-completeness conjecture for regular completions of trees

Let $T$ be a tree of maximum degree $\Delta$. For $d\geq\Delta$, let $\overline{T_d}$ be a $d$-regular graph obtained from $T$ by adding semi-edges or loops so that every vertex of $\overline{T_d}$ has degree $d$; this graph is not uniquely determined. Regular-tree completion conjecture. If $\Delta\geq 3$ and $\overline{T_d}$ is constructed as above, then the $\textsc{\overline{T_d}-Cover}$ problem is NP-complete even for simple input graphs. This is proposed as an open question arising from the paper's results, and its proof is described as a further goal.