Semi-regularity conjecture via the universal obstruction space

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $Y\subset X$ be a closed locally complete intersection subvariety of codimension $p$. Let $\mathrm{\mathbb{H}ilb}$ be the local Hilbert functor, let $O_{\mathrm{\mathbb{H}ilb}}$ be its universal obstruction space, and let

$$[v_e]:O_{\mathrm{\mathbb{H}ilb}}\to H^{1}(N_{Y/X})$$

be the natural linear monomorphism. Let $H^{1}(N_{Y/X})\to H^{p+1}(\Omega_{X/k}^{p-1})$ be the semi-regularity map. Bloch's semi-regularity conjecture. The composition of morphisms of $k$-vector spaces

$$O_{\mathrm{\mathbb{H}ilb}}\xrightarrow{[v_e]}H^{1}(N_{Y/X})\to H^{p+1}(\Omega_{X/k}^{p-1})$$

is trivial. This is an equivalent formulation of the assertion that the semi-regularity map annihilates all obstructions to embedded deformations of $Y$ in $X$.