Does the Witten genus determine $\mathrm{tmf}$ (or $\mathrm{TMF}$)?

Question

$\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.