Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ belong to $p$-fold Massey product of $ \tau_0$, i.e. $$ \langle \tau_0, \ldots, \tau_0 \rangle \ni \xi_1 ?$$
If so what is the proof or a reference?
If the result is false feel free to hammer down this question.
So far as definitions of Massey products, the dual Steenrod algebra is the homology $H_*(H\Bbb Z/p)$ of an $E_\infty$ ring spectrum, and so it has both Massey products and power operations.
Kraines, in Theorem 14 of "Massey higher products" from 1966, shows that the restricted power $\langle u \rangle^p \subset \langle u, u, \ldots, u\rangle$ can be identified with the power operation $-\beta P^m u$ when $u$ is an element in $H^{2m+1}(X; \Bbb Z/p)$. Kochman generalized this in "Symmetric Massey products and a Hirsch formula in homology" to certain differential graded Hopf algebras (of which the chains on the dual Steenrod algebra can be modeled by one, I think) showing that for such an algebra we have $\langle u \rangle^p = -\beta Q(u)$ plus a term involving iterated Browder brackets of $u$ and $\beta u$.
(In an ideal world, this would be an identification that's valid in the homology of any $E_\infty$ algebra, and not just one of these special forms.)
Then Theorem III.2.3 of Bruner-May-McClure-Steinberger's $H_\infty$ book proves that in the dual Steenrod algebra we have $\beta Q(\bar \tau_0) = \bar \xi_1$, and the conjugation in these degrees is thankfully just negation.
Thus $\langle \bar \tau_0,\ldots,\bar \tau_0\rangle = -\bar \xi_1$. This generalizes to $\tau_i$.
I learned this some years ago from Vigleik Angeltvelt (see his "Topological Hochschild homology and cohomology of $A_\infty$ ring spectra", Example 3.3).
