Deformation invariance of the Bounded Negativity Conjecture

Let $X$ and $Y$ be smooth complex projective surfaces that belong to a single family of complex projective surfaces.

Deformation-invariance conjecture. The Bounded Negativity Conjecture is true for $X$ if and only if it is true for $Y$.

If true, this would make bounded negativity a deformation-invariant property among smooth complex projective surfaces. The source gives no resolution evidence, so the claim remains open.