The truncated Anderson dual conjecture for real topological K-theory

Let $I_{\mathbb{Z}}$ denote the Anderson dual of the sphere spectrum, let ${}_4(I_{\mathbb{Z}})_{(2)}$ denote its relevant truncation after localization at $2$, and let $\operatorname{pic}_0^4(KO_{(2)})$ denote the fourth connective truncation of the Picard spectrum of real topological $K$-theory localized at $2$.

The truncated Anderson dual conjecture. There is an equivalence of spectra

$${}_4(I_{\mathbb{Z}})_{(2)}\simeq\operatorname{pic}_0^4(KO_{(2)}).$$

This is a 2-local, real analogue of the corresponding equivalence for complex topological $K$-theory. The homotopy groups of the two spectra agree in the calculations described, but the indeterminacy of the relevant $k$-invariants prevents a proof.