Existence conjecture for plane curves with prescribed graded Betti partition

Let $C:f=0$ be a reduced plane curve that is not free, with $t(C)=\sum_{j=1}^{m-2}\epsilon_j\geq 1$, and let $\pi_C$ be the ordered partition of $t(C)$ whose parts are the positive integers $\epsilon_j$. A curve has class $tT_{\pi}$ when $t(C)=t$ and $\pi_C=\pi$. Existence conjecture. For each $t\geq 1$ and each ordered partition $\pi$ of $t$, the set of plane curves of class $tT_{\pi}$ is not empty. This conjecture predicts that every ordered partition arising from the graded Betti numbers of a non-free reduced plane curve is realized by at least one plane curve; the supplied text gives no resolution or partial evidence.