The cusp connection conjecture for limit PK arrangements

Let $(L_j,\beta_j)$ be a weighted arrangement in $\mathbb{CP}^2$ satisfying all conditions of Theorem existence apart from one inequality, and suppose it has a multiple point $x$, called the cusp, such that

$$\sum_{j\mid x\in L_j}(\beta_j-1)=-2.$$

A limit PK arrangement with a cusp is such an arrangement. The cusp connection conjecture. For every limit $PK$ arrangement with a cusp there exists a flat torsion-free connection on $\mathbb{CP}^2$ with holonomy in the upper triangular subgroup of $SL(2,\mathbb{C})$ and with poles of residues $(0,\beta_j-1)$ at the lines $L_j$. This connection should preserve the sub-line bundle of $T\mathbb{CP}^2$ tangent to the pencil of lines through the cusp. This conjecture proposes a geometric structure for the simplicial arrangement series that are not $PK$ arrangements because the relevant inequality becomes an equality.