Taranenko's local-extrema conjecture for 1-polystochastic permanents

Taranenko’s conjecture asserts that every local maximum or minimum of the permanent on the relevant polytope occurs at a vertex or face centre. A paper dated April 2020 reports counterexamples, refuting the conjecture for even order and odd dimension.

April 2020 counterexamples

The paper constructs a positive-dimensional family of $1$-polystochastic matrices with permanent zero when $n$ is even and $d$ is odd. Because the permanent is nonnegative, every such matrix is a local minimum; for even $n>2$, these yield uncountably many local extrema outside the conjectured locations. The source explicitly concludes that the conjecture is false.

Current status (as of September 2026): The conjecture is reported false for even $n>2$ and odd $d$; no unresolved version or independent verification was found.