Irreducible bounded negativity conjecture

Let $X$ be a smooth projective surface, either rational or complex, and let $C$ be an effective reduced irreducible divisor on $X$. Irreducible bounded negativity conjecture. There is a bound $b_X$ such that

$$C^2\geq b_X.$$

This is presented as a further version of the Bounded Negativity Conjecture. The source gives no separate resolution status, so it remains open.