Kaplansky's zero-divisor conjecture

Claims a counterexample: a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space and nonzero $\alpha,\beta\in\mathbb F_2[G]$ with $\alpha\beta=0$. The group is the fundamental group of a cone complex over two random labelled graphs, with cancellation forced by odd label intersections; torsion-freeness comes from asphericity. It works in characteristic two only and does not disprove the zero-divisor conjecture over $\mathbb Q$ or $\mathbb C$, nor the Atiyah conjecture.

Claims a counterexample: a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space and nonzero $\alpha,\beta\in\mathbb F_2[G]$ with $\alpha\beta=0$. The group is the fundamental group of a cone complex over two random labelled graphs, with cancellation forced by odd label intersections; torsion-freeness comes from asphericity. It works in characteristic two only and does not disprove the zero-divisor conjecture over $\mathbb Q$ or $\mathbb C$, nor the Atiyah conjecture.