The bounded negativity conjecture for plane curves with bounded genus

Let $K$ be a fixed algebraically closed ground field. A multiplicity sequence is a positive integer sequence $(d,m_1,\ldots,m_r)$ arising from a reduced plane curve, with the $m_i$ recording singularity multiplicities, and a curve has multiplicity bound $m$ if $m_i\leq m$ for every $i$. A genus bound $g$ means that the normalization of each irreducible component has genus at most $g$. Bounded negativity conjecture for bounded genus. For every choice of integers $n\geq0$, $m>0$ and $g\geq0$, there are only finitely many multiplicity sequences arising for irreducible plane curves over $K$ with multiplicity bound $m$, genus bound $g$ and having at most $n$ singular points of multiplicity less than $m$.

This is one of the paper's proposed characteristic-free formulations related to bounded negativity. The source gives no resolution, so the conjecture remains open.