Singer's injectivity conjecture for the algebraic transfer

Let $\mathscr A$ be the Steenrod algebra over $\mathbb F_2$, and let $Tr_q(\mathbb F_2)$ denote the algebraic transfer in homology degree $q$.

Singer's conjecture. The homomorphism

$$Tr_q(\mathbb F_2)$$

is one-to-one for every $q$.

Singer's conjecture concerns injectivity of the algebraic transfer, a map relating coinvariants in the cohomology of elementary abelian 2-groups to $\operatorname{Ext}$ over the Steenrod algebra. It is known in low homological degrees, including $q=3$ and $q=4$, but the paper states that it fails in bidegree $(6,6+36)$, so the conjecture is refuted.