The canonical-degree linear bound conjecture for curves on smooth projective surfaces

Let $X$ be a smooth complex projective surface. For an integral curve $C$ on $X$, let $g=g(C)$ be its geometric genus and let $k_C:=C.K_X$ be its canonical degree.

Canonical-degree bound conjecture. There exist constants $A$ and $B$, depending only on $X$, such that for every integral curve $C$ on $X$,

$$k_C\leqslant A(g-1)+B.$$

This conjecture is widely open. The paper proves the bound with $A=-1$ under the additional assumptions $h^0(X,-K_X)=0$ and $h^0(X,2K_X+C)=0$.