Existence conjecture for curves with Jacobian syzygies of the same degree

Let $C:f=0$ be a reduced plane curve of degree $d$, and let $d_1\leq\cdots\leq d_m$ be the degrees of its minimal Jacobian syzygies, with $r=d_1$. A curve of type $(d,r,m)$ is one satisfying

$$r=d_1=d_2=\cdots=d_m.$$

Existence conjecture for curves of type $(d,r,m)$. For any integers $d,m\geq 3$ and any integer $r$ such that

$$\frac{d}{2}\leq r\leq d-1\quad\text{and}\quad m<2r-d+3,$$

there are curves of type $(d,r,m)$.

The conjecture proposes existence throughout the indicated range for curves whose Jacobian syzygies have the same degree but whose number of syzygies is below the maximal-Tjurina value. The source presents examples in many cases, but does not state that the general assertion has been proved, so the conjecture remains open.