Pseudoline arrangement conjecture for the Hirzebruch quadratic form

Let $\mathcal{L}\subset\mathbb{RP}^2$ be an essential pseudoline arrangement, and let $M$ be its matroid. Let $Q$ be the Hirzebruch quadratic form of $M$, and let the semistable cone and its interior be those associated with $M$. Pseudoline arrangement conjecture. The form $Q$ is non-positive on the semistable cone of $M$; moreover, whenever $Q$ vanishes in the interior of that cone, $M$ is realizable over $\mathbb{R}$ and the pseudoline arrangement $\mathcal{L}$ is stretchable. This is proposed as a possible extension of the theorem for realizable arrangements to non-realizable matroids; the source describes it as provocative and possibly over-optimistic, and gives no evidence of a resolution.