Regularity and arithmetically Cohen–Macaulay conjecture for secant varieties of graph curves

Let $C_G$ be a graph curve embedded under the hypotheses of Theorem 1.3, and let $k$th secant variety denote the union of the linear spans of length-$k+1$ subschemes of $C_G$. Secant-variety regularity conjecture. Under the hypotheses of Theorem 1.3, the $k$th secant variety of $C_G$ has regularity equal to $2k+1$ and is arithmetically Cohen–Macaulay. This is presented as the graph-curve analogue of a conjecture for secant varieties of smooth curves; the source does not give a resolution status.