Odd-primary characteristic conjecture for $S/\!/p^r,\alpha_1$

Let $p$ be an odd prime and let $r>1$. The commutative $S$-algebra $S/\!/p^r,\alpha_1$ is obtained by attaching an $\mathcal{E}_\infty$ cell to $S/\!/p^r$ to kill the surviving element $\alpha_1\in\pi_{2p-3}(S/\!/p^r)$. A commutative $S$-algebra $R$ is a characteristic for a ring spectrum $A$ when its unit induces a map on homotopy groups whose kernel contains the kernel of the unit $S\to A$.

Odd-primary characteristic conjecture. $S/\!/p^r,\alpha_1$ is a characteristic for $H\mathbb{Z}/p^r$.

The preceding construction shows that attaching one more cell to kill the class $u'_1$ produces a wedge of suspensions of Eilenberg–Mac Lane spectra $H\mathbb{Z}/p^s$ for $0\leqslant s\leqslant r$, but it does not establish the asserted characteristic property.