The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide

Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{G}\mathrm{L}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash–Miller–Patzt, uses the sharbly resolution. The other, by Brown–Chan–Galatius–Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.