  Writing $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra
of stable natural endomorphisms of the mod 2 cohomology functor on topological
spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an
important problem of Algebraic topology, because it is the initial page of the
Adams spectral sequence converging to stable homotopy groups of the spheres. A
relatively efficient tool to describe this cohomology is the Singer algebraic
transfer of rank $n$ in [Math. Z. 202 (1989), 493-523], which passes from a
certain subquotient of a divided power algebra to the cohomology of $\mathbb
A.$ Singer predicted that this transfer is a monomorphism, but this remains
open for $n\geq 4.$ This short note is to verify the conjecture in the ranks 4
and 5 and some generic degrees.
