Secant-plane nonexistence conjecture for graph curves

Let $C_G\subset\mathbb{P}^{g+r}$ be a graph curve of arithmetic genus $p_a$ and degree $d=2g+r$. A $(k+2)$-secant $\mathbb{P}^k$ is a projective $k$-plane meeting $C_G$ in a subscheme of length at least $k+2$. Secant-plane nonexistence conjecture. As long as $2k+1<g+r$, $C_G$ has no $(k+2)$-secant $\mathbb{P}^k$. The conjecture is motivated by the expected behavior of secant varieties in smoothings of graph curves and by the claim that the relevant secant variety should fill the ambient projective space before such collapsing can occur.