Tohăneanu's low-exponent supersolvability conjecture for free arrangements

Let $\mathcal{A}\subset\mathbb P^{k-1}$ be a free hyperplane arrangement with exponents

$$\operatorname{exp}(\mathcal{A})=(1,\underbrace{2,\ldots,2}_{k-2},3).$$

Tohăneanu's low-exponent supersolvability conjecture. Then $\mathcal{A}$ is supersolvable. The conjecture concerns free arrangements with exactly one exponent equal to $3$ and is verified in ranks $4$ and $5$, as well as for inductively free arrangements of arbitrary rank; the general case remains open.