The free Picard n-groupoid conjecture for the sphere spectrum

Let $\mathbb{S}=\Sigma^\infty S^0$ be the sphere spectrum, and let $\tau_{[0,n]}\mathbb{S}$ denote its truncation to degrees from $0$ through $n$. Let $\mathcal{F}_L^{(n)}$ be the free Picard $n$-groupoid on one formal object $L$. Free Picard n-groupoid conjecture. The Picard $n$-groupoid corresponding to $\tau_{[0,n]}\mathbb{S}$ should be identified with $\mathcal{F}_L^{(n)}$. This is part of the proposed dictionary between spectra and Picard $n$-groupoids, with the sphere spectrum playing the role of the integers as the unit of the symmetric monoidal stable homotopy category. The supplied text does not indicate whether the conjecture has been proved or disproved.