Irreducibility conjecture for extremal logarithmic 2-jet differentials

Let $cmathcal{C}subsetcmathbb{P}^2$ be a configuration of three conics with simple normal crossings. For any $cepsilon>0$, define the logarithmic $2$-jet differential space by

$$H^0\!\left(\mathbb{P}^2,E_{2,m}T^*_{\mathbb{P}^2}(\operatorname{log}\mathcal{C})\otimes\mathcal{O}_{\mathbb{P}^2}(-t)\right).$$

Irreducibility conjecture. For every $cepsilon>0$, there exists a nonzero negatively twisted invariant logarithmic $2$-jet differential $comega$ in this space with

$$\overline{\theta}_2(X,\mathcal{C})-\epsilon<\frac{t}{m}<\overline{\theta}_2(X,\mathcal{C}),$$

whose zero locus $c{\omega=0}\subset X_2$ is irreducible and reduced modulo $cGamma_2$. This conjecture would avoid the key vanishing lemmas and yield a Second Main Theorem constant approximately $1/\overline{\theta}_2(\mathbb{P}^2,\mathcal{C})<3.58\ldots$; its resolution is not supplied here.