Let $H$ be the Hilbert scheme of Artin local rings (quotients of a power series ring $R$ in $e$ variables over $\mathbb{C}$) of length $n$. Consider the set $G\subset H$ of rings $A$ with the property that the kernel of $d:A\to \Omega^1_A$ is just the constants. Is $G$ dense in $H$?

This is true if $e\leq 2$, thanks to the irreducibility of $H$. In general, I have no idea and pointers to relevant information would be appreciated.