The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus I am aware of a lift to a "spin orientation of Tate K-theory", namely to a map of ring spectra $M \mathrm{Spin}\to KO[ [q] ]$: lemma 5.4, 5.8 in

• Matthias Kreck, Stefan Stolz, $HP^2$-bundles and elliptic homology, Acta Math, 171 (1993) 231-261

At least naively one would hope to see a further refinement of the Ochanine genus to an orientation of something like $tmf_0(2)$. Is there some reason why this should not work, or has it just not been considered?