The naive equivariant ring-spectrum conjecture for motivic modular forms

Let $C_2$ be the cyclic group of order two, and let $\mathrm{tmf}_{C_2}$ be the homotopy pullback spectrum defined by the Tate diagram described above. A $C_2$-ring spectrum means a ring spectrum equipped with a naive $C_2$-equivariant structure. The naive equivariant ring-spectrum conjecture. The $C_2$-spectrum $\mathrm{tmf}_{C_2}$ is a $C_2$-ring spectrum in the naive sense. The conjecture asserts that the proposed equivariant refinement of $\mathrm{tmf}$ admits the expected naive equivariant multiplicative structure; the surrounding discussion explains that this would follow if the rightmost map in the defining pullback diagram were a map of ring spectra.