Greuel–Lossen–Shustin asymptotic properness conjecture for the gamma invariant

Let $V_d^{\operatorname{irr}}(S_1,\dots,S_r)$ denote the family of reduced irreducible complex plane curves of degree $d$ with isolated singular points of prescribed topological types. For a topological singularity $S$, let $\gamma(S)$ be the singularity invariant used in the smoothness criterion, and let $V_d^{\operatorname{irr}}(r\cdot S)$ denote the corresponding family with $r$ singularities of type $S$. Asymptotic properness conjecture. There exists an absolute constant $A>0$ such that, for every topological singularity $S$, there are infinitely many pairs $(r,d)\in\mathbb{N}^2$ for which $V_d^{\operatorname{irr}}(r\cdot S)$ is empty, or is not smooth, or has dimension greater than the expected dimension, and

$$r\cdot\gamma(S)\leq A\cdot d^2.$$

The paper proves that $\sum_i\gamma(S_i)<d^2+6d+8$ suffices for smoothness and expected dimension, and conjectures that the exponent and the invariant $\gamma$ are asymptotically optimal for topological singularities.