Asymptotic existence of nodal curve components with bounded numbers of nodes

For integers $r,d,g,x$ with $0\le x\le g$, let $E'(r,d,g,x)$ be the locus of integral, non-degenerate degree-$d$ curves in $\mathbb{P}^r$ of arithmetic genus $g$ having exactly $x$ ordinary nodes as their only singularities and satisfying $h^1(C,\mathcal{O}_C(2))=0$. An irreducible component $\Gamma$ of this locus has expected dimension when its Hilbert-scheme dimension is $(r+1)d-(r-3)(g-1)$, and has the expected number of moduli when

$$\dim\bigl(u_{r,d,g,x}(\Gamma)\bigr)=\min\{3g-3-x,3g-3+\rho(d,g,r)-x\}.$$

Bounded-nodes existence conjecture. Fix integers $r\geq 3$ and $t\geq 0$. There is a function $\mathbb{O}_{r,t}:\mathbb{N}\to\mathbb{N}$ such that $\lim_{g\to+\infty}\mathbb{O}_{r,t}(g)/g=0$ and an integer $g_0\geq t$ such that, for all integers $g,x,d$ with $g\geq g_0$, $0\leq x\leq t$, and $-\rho(g,r,d)+x\leq 3g-\mathbb{O}_{r,t}(g)$, the locus $E'(r,d,g,x)$ has an irreducible component with expected dimension and the expected number of moduli.

This is a stronger asymptotic assertion in the bounded-node regime, allowing every number of nodes up to a fixed $t$ while retaining the expected-dimensional and expected-moduli conclusions. The source supplies no evidence that the conjecture has been resolved.