The structured ring spectrum conjecture for $\Mell^\dR$ and $\TMF/\!\/\nu$

Let $\mathcal{M}_\mathrm{ell}^\mathrm{dR}$ be the moduli stack of elliptic curves equipped with a splitting of the Hodge filtration, and let $\mathcal{M}_\mathrm{ell}$ and $\mathcal{M}_{FG}$ denote the moduli stacks of elliptic curves and formal groups, respectively. For some $k\geq 2$, a sheaf of even-periodic \mathbf{E}_{{_}}k-rings $\mathcal{O}^\mathrm{der}$ should exist on the étale site of $\mathcal{M}_\mathrm{ell}^\mathrm{dR}$ such that, for every étale map $f:\operatorname{Spec} R\to \mathcal{M}_\mathrm{ell}^\mathrm{dR}$, the value $\mathcal{O}^\mathrm{der}(f)$ is the Landweber-exact theory associated to the composite

$$\operatorname{Spec} R\to \mathcal{M}_\mathrm{ell}^\mathrm{dR}\to \mathcal{M}_\mathrm{ell}\to \mathcal{M}_{FG}.$$

Structured ring spectrum conjecture. The global sections satisfy

$$\Gamma(\mathcal{M}_\mathrm{ell}^\mathrm{dR};\mathcal{O}^\mathrm{der})\simeq \mathrm{TMF}/\!\/\nu$$

as \mathbf{E}_{{_}}1-rings, and the resulting \mathbf{E}_{{_}}k-ring structure on $\mathrm{TMF}/\!\/\nu$ extends to an \mathbf{E}_{{_}}k-ring structure on $\mathrm{tmf}/\!\/\nu$. The conjecture proposes a sheaf-level refinement of the descent spectral sequence for $\mathrm{tmf}/\!\/\nu$; its status is not resolved in the supplied source.