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$\newcommand\specfont[1]{\mathrm{#1}}$$\newcommand\MSpin{\specfont{MSpin}}\newcommand\KO{\specfont{KO}}\newcommand\KU{\specfont{KU}}\newcommand\MString{\specfont{MString}}\newcommand\tmf{\specfont{tmf}}\newcommand\TMF{\specfont{TMF}}\newcommand\pt{\mathrm{pt}}\newcommand\isoarrow{\overset\sim\longrightarrow}$It is shown in Spin cobordism determines real K-theory, by Hopkins and Hovey, that the Atiyah–Bott–Shapiro orientations $\MSpin \to \KO$ and $\MSpin^c\to \KU$ determine $\KO$ and $\KU$ via their associated genera by Landweber construction, i.e., that one has natural isomorphisms $$ \MSpin(X)\otimes_{\MSpin(\pt)}\KO(\pt) \isoarrow \KO(X) $$ and $$ \MSpin^c(X)\otimes_{\MSpin^c(\pt)}\KU(\pt) \isoarrow \KU(X). $$ One may wonder whether the same holds for the Ando–Hopkins–Rezk orientation $\MString \to \tmf$, i.e., whether the Witten genus induces a natural isomorphism $$ \MString(X)\otimes_{\MString(\pt)}\tmf(\pt) \isoarrow \tmf(X) $$ or whether it possibly does so after passing to the periodic version $\TMF$ of $\tmf$, i.e., whether it induces a natural isomorphism $$ \MString(X)\otimes_{\MString(\pt)}\TMF(pt) \isoarrow \TMF(X). $$ The question, in slightly more vague terms as the Ando–Hopkins–Rezk orientation was not yet known, is already present in the Hopkins–Hovey paper. It is likely an answer is by now known, either in the positive or in the negative, but I have so far been unable to locate it in the literature. As indicated in the comment by Eric Peterson, the periodic version of the question appears as an open question under investigation in a set of notes by Sanath Devalapurkar from the year 2020.

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    $\begingroup$ I think this is unknown. Gerd Laures once gave a(n unrecorded) related talk based off of arxiv.org/abs/1710.03427 . He summarized the preceding cases (yours, MO –> HF2, MSO –> HZ) as: analyze the ordinary cohomology of the bordism spectrum, find the submodule supported by the unit class, and apply a Milnor–Moore-type theorem. He said his recourse to K(2)-local homotopy was necessary because H^* MString does not split cleanly as an A // A(2)-module, which stymies the traditional approach — though I do not know exactly what’s unclean nor the ultimate fate of this base-change theorem. $\endgroup$ Oct 25, 2021 at 18:51
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    $\begingroup$ One more comment: Sanath Devalapurkar is actively working on results in this area, and your question appears as still open in these slides from last year: sanathdevalapurkar.github.io/files/… . $\endgroup$ Oct 26, 2021 at 14:11
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    $\begingroup$ I don't think that this can be true with the connective version of $\text{tmf}$, although I don't immediately see a proof; certainly that would be a very different result from the $K$-theory results that you cite. It is much more likely to be true for periodic $\text{TMF}$. $\endgroup$ Oct 26, 2021 at 21:22
  • $\begingroup$ Thanks. I'm now editing the question so to include the periodic version. $\endgroup$ Oct 26, 2021 at 21:46

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