[1][2][3] A later preprint by Gardam claims that essentially the same element also gives a counter-example in characteristic 0 (finding an inverse is computationally much more involved in this setting, hence the delay between the first result and the second one).
[5] It follows that the conjecture holds more generally for all residually torsion-free elementary amenable groups.
The unit conjecture is also known to hold in many groups, but its partial solutions are much less robust than the other two (as witnessed by the earlier-mentioned counter-example).
This conjecture is not known to follow from any analytic statement like the other two, and so the cases where it is known to hold have all been established via a direct combinatorial approach involving the so-called unique products property.
In the mid-1970s, H. Garth Dales and J. Esterle independently proved that, if one furthermore assumes the validity of the continuum hypothesis, there exist compact Hausdorff spaces X and discontinuous homomorphisms from C(X) to some Banach algebra, giving counterexamples to the conjecture.
In 1976, R. M. Solovay (building on work of H. Woodin) exhibited a model of ZFC (Zermelo–Fraenkel set theory + axiom of choice) in which Kaplansky's conjecture is true.
In 1953, Kaplansky proposed the conjecture that finite values of u-invariants can only be powers of 2.