Hypersurface Convexity and Extension of Kähler Forms

The following generalization of a result of S. Nemirovski is proved: if $X$ is either a projective or a Stein manifold and $K\subset X$ is a compact sublevel set of a strictly plurisubharmonic function $\phi$ defined in a neighborhood of $K$, then $X\setminus K$ is a union of positive divisors if and only if $d{d}^c\phi$ extends to a Hodge form on $X$. For an arbitrary compact subset $K⊊X$, this gives that $X\setminus K$ is a union of positive divisors if and only if $K$ admits a neighbourhood basis of sublevel sets of strictly plurisubharmonic functions with the $d{d}^c$-extension property.