The sphere-spectrum V-completion conjecture

Let $\operatorname{Set}^{\mathrm{fin}}_{\cong}$ be the category of finite sets and bijections, let $\widetilde{K}(\operatorname{Set}^{\mathrm{fin}}_{\cong})$ denote its reduced algebraic $K$-theory spectrum with Verschiebung filtration, and let $S$ denote the sphere spectrum. The sphere-spectrum V-completion conjecture. There is a weak equivalence of spectra

$$\operatorname{holim}_j\,\widetilde{K}(\operatorname{Set}^{\mathrm{fin}}_{\cong})/V^j\simeq \operatorname{TR}(S),$$

lifting the isomorphism

$$\varprojlim_j K_0(\operatorname{Set}^{\mathrm{fin}})/V^j\longrightarrow W(\mathbb{Z}).$$

This is presented as an easier related conjecture, and remains unproved in the source.