Lifting the free δ_p-ring on one generator to the sphere

Determine whether the free δ_p-ring on one generator admits a realization as a commutative ring spectrum Moore spectrum; that is, decide whether there exists an E_infinity-ring spectrum Sph_R with π_0(Sph_R) the free δ_p-ring on one generator and the induced Frobenius lift agreeing with its δ_p-structure.

Background

A central theme of the paper is the lifting problem: for which widehat-delta rings R can one construct a commutative ring spectrum Sph_R with π_0 = R and compatible Frobenius operations, and when do mapping spaces between such lifts coincide with morphisms of widehat-delta rings? Known examples include spherical Witt vectors, but few other E_infinity lifts of Moore spectra are known.

The authors single out a basic test case: the free δ_p-ring on one generator. Establishing its liftability would provide a new example of a spherical lift in the E_infinity setting and inform the broader program connecting δ-structures and highly structured multiplicative Moore spectra.

References

As for the first part of \Cref{question}, the lifting question is equivalent to the question of highly structured multiplications on Moore spectra and has been extensively studied , though we do not know many interesting examples of spherical lifts in the $\E_{\infty}$ case beyond the spherical Witt vectors of Example 5.2.7. For instance, we believe the following is open:

\begin{question} Does the free $\delta_p$-ring on one generator lift to $\Sph$? \end{question}

— Maps between spherical group rings (2405.06448 - Carmeli et al., 10 May 2024) in Introduction, Background and motivation (Question on lifting δ_p-rings)