Brylinski's positive-energy representation conjecture for elliptic cohomology

Let $m$ be an integer, and let $P_m(d)$ be the free abelian group generated by the isomorphism classes of irreducible positive-energy representations of the universal central extension $\tilde{L}\operatorname{Spin}(d)$ of the free loop group $L\operatorname{Spin}(d)$ at level $m$. Let $B\operatorname{String}(d)$ denote the 7-connected cover of $B\operatorname{Spin}(d)$, and let $\lambda$ be the elliptic character. Brylinski's conjecture. There is an integer $n$ depending on $d$ and $m$ and an additive map

$$\varphi_d:P_m(d)\longrightarrow TMF(n)^0B\operatorname{String}(d)$$

such that, for every $V\in P_m(d)$, its elliptic character agrees with the associated bundle

$$\lambda\varphi_d(V)=(\tilde{L}E\operatorname{Spin}(d)\times_{\tilde{L}\operatorname{Spin}(d)}V)_{|B\operatorname{String}(d)}.$$

This conjectures that positive-energy loop-group representations define classes in topological modular forms whose elliptic characters recover the expected associated bundles. The source attributes the underlying idea to Brylinski; the supplied text does not state whether the conjecture has been proved or disproved.