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New tests are applied to two $\beta$-functions of the much-discussed BSMmodel with ten massless fermion flavors in the fundamental representation ofthe SU(3) color gauge group. The renormalization scheme of the two$\beta$-functions is defined on the gauge field gradient flow in respectivefinite or infinite physical volumes at zero lattice spacing. Recently publishedresults in the ten-flavor theory led to indicators of an infrared fixed point(IRFP) in the finite-volume step $\beta$-function in the strong coupling regimeof the theory arXiv:2004.00754. We analyze our substantially extended set often-flavor lattice ensembles at strong renormalized gauge couplings and find noevidence or hint for IRFP in the finite-volume step $\beta$-function withincontrolled lattice reach. We also discuss new ten-flavor tests of the recentlyintroduced lattice definition and algorithmic implementation of the$\beta$-function defined on the gradient flow of the gauge field over infiniteEuclidean space-time in the continuum. Originally we introduced this newalgorithm to match finite-volume step $\beta$-functions in masslessnear-conformal gauge theories with the infinite-volume $\beta$-function reachedin the chiral limit from small fermion mass deformations of spontaneous chiralsymmetry breaking.  Results from the lattice analysis of the ten-flavor infinite-volume$\beta$-function are consistent with the absence of IRFP from our step$\beta$-function based analysis. We make important contact at weak coupling ininfinite volume with gradient flow based three-loop perturbation theory,serving as a first pilot study toward the long-term goal of developingalternate approach to the determination of the strong coupling $\alpha_s$ atthe Z-boson pole in QCD.

From ten-flavor tests of the $\beta$-function to $\alpha_s$ at the Z-pole