Existence conjecture for hypersurfaces with prescribed isolated singularities

Let $X$ be a projective algebraic variety of dimension $n$, let $W$ be a very ample linear system on $X$, and let $S_1,\dots,S_r$ be singularity types with Milnor numbers $\mu(S_i)$. Existence conjecture for prescribed singularities. There exists a constant $A=A(X,W)>0$ such that, for each collection $S_1,\dots,S_r$ of singularity types and for each positive integer $d$ satisfying

$$\sum_{i=1}^r\mu(S_i)<Ad^n,$$

a hypersurface $W_d\in|dW|$ exists with exactly $r$ isolated singularities of types $S_1,\dots,S_r$, respectively. This is the expected higher-dimensional analogue of the existence results for curves with prescribed singularities; the source presents it as an open problem for hypersurfaces of dimension greater than one, where the discussion is restricted to analytic types.