Characteristic conjecture for $S/\!/2^r,\eta$

Let $r>1$. The commutative $S$-algebra $S/\!/2^r,\eta$ is obtained from $S/\!/2^r$ by killing the image of $\eta$. 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 $S\to A$.

Characteristic conjecture. $S/\!/2^r,\eta$ is a characteristic for $H\mathbb{Z}/2^r$.

The related object $S/\!/2^r,\eta,u'_1$ splits as a wedge of suspensions of Eilenberg–Mac Lane spectra $H\mathbb{Z}/2^s$ for $0\leqslant s\leqslant r$. The authors explicitly warn that they have no direct evidence for this conjecture and that it may be too optimistic.