Existence conjecture for maximal Tjurina curves and line arrangements

Let $C:f=0$ be a reduced plane curve of degree $d$, let $r={\rm mdr}(f)$, and suppose $d/2\leq r\leq d-1$. A maximal Tjurina curve of type $(d,r)$ is one satisfying

$$\tau(C)=(d-1)^2-r(d-r-1)-\binom{2r+2-d}{2}.$$

Maximal Tjurina existence conjecture. For every integer $d\geq 3$ and every integer $r$ with $d/2\leq r\leq d-1$, there are maximal Tjurina curves of type $(d,r)$. Moreover, for $d/2\leq r\leq d-2$, there are maximal Tjurina line arrangements of type $(d,r)$.

The conjecture is known for many pairs $(d,r)$, in particular for all relevant pairs with $d\leq 11$, but the general case remains open. The constructions include line arrangements and irreducible curves with notable geometry, such as maximal nodal curves when $r=d-1$.