Yoshihara's conjecture on complements of irreducible plane curves

Let $\mathrm{k}$ be an algebraically closed field of arbitrary characteristic, and let $C,D\subset {\mathbb P}^2$ be irreducible curves. Two such curves are projectively equivalent if an automorphism of ${\mathbb P}^2$ sends $C$ to $D$. Suppose that

$$\varphi\colon {\mathbb P}^2\setminus C\longrightarrow {\mathbb P}^2\setminus D$$

is an isomorphism between their complements.

Yoshihara's conjecture. Then $C$ and $D$ are projectively equivalent.

The conjecture asserts that the complement of an irreducible plane curve determines the curve up to projective equivalence. It is refuted: a counterexample was given by Blanc in 2009.