The higher-prime Hopf-algebra conjecture for eo_{p-1}

Let $p$ be a prime, let $\operatorname{eo}_{p-1}$ be the proposed connective form of $EO_{p-1}$, and let $a_2,\dots,a_{p-1}$ be exterior generators with

$$|a_i|=2i(p-1)+1.$$

Let $\operatorname{\mathcal A}(1)$ be dual to the subalgebra generated by $\beta$ and $\mathcal P^1$, with its usual coproducts; set $a_1=\tau_1$ and $a_0=\tau_0$. The higher-prime Hopf-algebra conjecture. As a Hopf algebra,

$$\pi_*\big(H\operatorname{\mathbb Z}/p\wedge_{\operatorname{eo}_{p-1}}H\operatorname{\mathbb Z}/p\big)=\operatorname{\mathcal A}(1)\otimes E(a_2,\dots,a_{p-1}),$$

with coproducts

$$\psi(a_j)=\sum_{k=0}^{j}\frac{1}{k!}\xi_1^k\otimes a_{j-k}+a_j\otimes 1.$$

The spectra $\operatorname{eo}_{p-1}$ are not known to exist for $p>3$, so this follows a heuristic pattern from $\operatorname{eo}_2$; after inverting $\Delta$, the corresponding Adams spectral sequence is expected to recover the homotopy of $EO_{p-1}$.