Greuel–Lossen–Shustin conjecture for general weighted plane configurations

Let $({\mathcal C},\boldsymbol{r})$ be a consistent finite weighted configuration of infinitely near points of $\mathbb{P}^2$, where $r_p$ is the multiplicity assigned to $p$, and let

$${\mathcal H}_{({\mathcal C},\boldsymbol{r})}=\pi_*{\mathcal O}_X\left(-\sum_{p\in{\mathcal C}}r_pE_p^*\right)$$

be the associated ideal sheaf. Here $u$ ranges over configurations general among those with the same proximities, and $d$ is larger than the \sum of the three largest multiplicities. Greuel–Lossen–Shustin conjecture. If $k=\mathbb{C}$, then

$$h^0\left(\mathbb{P}^2,{\mathcal O}_{\mathbb{P}^2}(d)\otimes{\mathcal H}_{({\mathcal C},\boldsymbol{r})}\right)=\max\left\{0,\frac{(d+1)(d+2)}{2}-\sum_{p\in{\mathcal C}}\frac{r_p(r_p+1)}{2}\right\}.$$

The conjecture asserts the expected dimension for sections of general plane configuration ideals and is used in the paper to derive the Nagata-type minimality conjecture for very general valuations.