Let $Y$ be a nonsingular variety and $X\subset Y$ a closed subscheme. A *linear scheme* over $X$ is a scheme of the form $\textbf{Spec}\, ( Sym _{\mathcal{O}_X} F) $, where $F$ is a coherent sheaf on $X$.

If the embedding $X\subset Y$ is regular, i.e. if every point of $Y$ has a neighborhood over which the ideal $I$ defining the above embedding is generated by a regular sequence, then it is well known that the normal cone $C_{X/Y}= \textbf{Spec}\, (\oplus _{i\geq 0} I^i/I^{i+1})$ is isomorphic to the linear scheme $ \textbf{Spec}\, ( Sym _{\mathcal{O}_X} I/I^2) $- the total space of the conormal sheaf.

**Question**: Is the converse true? That is, suppose that $X$ is a closed subscheme of a nonsingular variety $Y$ and $C_{X/Y}$ is isomorphic to a linear scheme. Is it true that $X\subset Y$ is regular embedding?