Let $X$ be a compact Kähler manifold and $S \subset X$ a closed complex submanifold. Given a closed $(1, 1)$-form $\alpha$ on $S$, is there always a closed $(1, 1)$-form $\beta$ on a neighborhood of $S$ in $X$ such that $[\beta|_S] = [\alpha]$ in $H^{1, 1}(S, \mathbb{C})$?

Note. This mathoverflow answer seems related, but I don't think it answers the above question. The proof of Theorem 4.1 in arXiv:math/0609617 begins by assuming that we have such an extension and pursue by proving positivity results. I fail to see how any of the ideas in the proof would help with the question above. But I might be wrong, so any help in that direction would be great too.