Nonrealizability conjecture for supersolvable arrangements with a triple-point line

Let $\mathcal A\subset\mathbb P^2$ be a full-rank supersolvable arrangement of $n$ lines with a singular modular point $P$ of maximum multiplicity $m\geq 3$. A line has only triple singularities when every singular point on it has multiplicity $3$. Nonrealizability conjecture. If there exists a line $\ell\in\mathcal A$ not passing through $P$ and having only triple singularities, then $\mathcal A$ is not realizable over $\mathbb C$.

This is proposed as a route toward the complex supersolvable Dirac–Motzkin conjecture: establishing the nonrealizability assertion would rule out the configurations needed for a line away from the modular point to have no simple points. The paper does not prove the assertion in general.