Linear weighted bounded negativity conjecture

Let $X$ be a smooth projective surface over an algebraically closed field of characteristic zero. Let $C$ be a reduced and irreducible curve on $X$, and let $D$ be a big and nef divisor on $X$ such that $D\cdot C>0$. Linear weighted bounded negativity conjecture. There exists a non-negative integer $B_W(X)$, depending only on $X$, such that

$$\frac{C^2}{D\cdot C}\geq -B_W(X)$$

for all such curves $C$ and divisors $D$. This conjecture proposes a linear bound on the self-intersection $C^2$ in terms of $D\cdot C$, strengthening the quadratic bound predicted by the weighted bounded negativity conjecture; its status is open.