The degree-uniform bounded negativity conjecture for smooth surfaces in projective three-space

Fix an integer $d\geq1$. A smooth surface of degree $d$ in $\mathbb{P}^3$ is a smooth projective surface embedded in $\mathbb{P}^3$ by a homogeneous equation of degree $d$; its negativity bound is a uniform lower bound for self-intersection numbers of its curves.

Degree-uniform bounded negativity conjecture. There exists a constant $c_d$, depending only on $d$, such that every smooth surface of degree $d$ in $\mathbb{P}^3$ has negativity bound $c_d$.

This conjecture seeks a bound uniform across the connected family of smooth degree-$d$ surfaces, rather than a bound chosen separately for each surface. The source gives no resolution evidence.