Expected intersection-point conjecture for two plane curves

Let $D$ be a general degree-$d$ plane curve, and let $i(d,m)$ be the minimal number of points in the set-theoretical intersection $C\cap D$ as $C$ ranges over irreducible degree-$m$ curves meeting $D$ properly. For $m<d$, set

$$r_{d,m}=\frac{m(m+3)}{2}.$$

Expected intersection-point conjecture. For $d>m$ and $d\ge 3$,

$$i(d,m)=dm-r_{d,m}.$$

The paper explains that the expected formula is false when $m\ge d\ge3$, proves the conjecture when $2d\ge3m-2$ (in particular for $m\le4$), and leaves the remaining cases open.