Invariant two-jets and effective hyperbolicity for complements of two plane curves

Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple normal crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem for a general ordered pair whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{o