The status of this question in OPEN.
This theory has NOT been developed yet.
That being said, the evidence is as compelling as ever, I don't know of any obstructions to making this work, and I'm convinced that there's an awesome theory out there, waiting to be discovered.
Roughly 8 years ago, I wrote an unsuccessful ERC proposal, where I outlined a program. This proposal can be found on my Utrecht website here and here (warning: that website will probably stop existing one year from now, so the links will become broken -- but the linked material will still be there to be found on whatever new website I end up having in the future).
There are small bits and pieces of what one might call progress, which I've made available on my website:
Here's one. In this draft, I take a compact simply connected Lie group $G$ of dimension $d$, and I consider the map $p:G\to \{pt\}$. I construct, geometrically, the $TMF$-pushforward $p_!(1)\in TMF^{-d}(\{pt\})=\pi_d(TMF)$ of the element $1\in TMF^0(G)$ along the map $G\to \{pt\}$.
Here's another one. In this draft, I show that there's a new type of 2-equivariance for $TMF$, where the group of equivariance gets replaced by a fusion category.
