Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three

Let $\mathcal{M}$ be a simple matroid of rank $r$ on the ground set $[n]=\{1,\dots ,n\}$, let $T_{\mathcal{M}}(x,y)$ denote its Tutte polynomial, and let $\mathrm{O}\mathrm{S}(\mathcal{M})$ denote its Orlik-Solomon $\mathbb{C}$-algebra. We say that $\mathrm{O}\mathrm{S}(\mathcal{M})$ determines $T_{\mathcal{M}}(x,y)$ if, for every simple matroid $\mathcal{N}$, an isomorphism $\mathrm{O}\mathrm{S}(\mathcal{M})\cong \mathrm{O}\mathrm{S}(\mathcal{N})$ of $\mathbb{C}$-algebras, without regard to the grading, implies $T_{\mathcal{M}}(x,y)=T_{\mathcal{N}}(x,y)$. Falk conjectured that the Orlik-Solomon $\mathbb{C}$-algebra of every simple matroid of rank three determines its Tutte polynomial. We prove that this holds for every simple matroid of rank at most three, thereby resolving Falk’s conjecture.