The spherical framed Floer spectrum identification conjecture

Let $\mathfrak{R}^{\mathrm{sfr}}$ be the spherically framed endomorphism ring spectrum, let $BJ$ denote the 1-fold delooping of the $J$-homomorphism, and let $\mathrm{hofib}(BJ)$ be its homotopy fiber.

Spherical framed Floer spectrum identification conjecture. There is a homotopy equivalence of ring spectra

$$\mathfrak{R}^{\mathrm{sfr}}\simeq\Sigma^\infty\mathrm{hofib}(BJ).$$

This identifies the spherically framed Floer endomorphism spectrum with a suspension spectrum and is a proposed structural description related to the comparison with spherical framed bordism. The supplied text gives no evidence that this identification has been resolved.