New tests are applied to two $\beta$-functions of the much-discussed BSM model with ten
massless fermion flavors in the fundamental representation of the SU(3) color gauge group.
The renormalization scheme of the two $\beta$-functions is defined on the gauge field gradient flow in respective finite or infinite physical volumes at zero lattice spacing. Recently published results 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 regime of the theory [1]. We analyze our substantially extended set of ten-flavor lattice ensembles at strong renormalized gauge couplings and find no evidence or hint for IRFP in the finite-volume step $\beta$-function within controlled lattice reach. We also discuss new ten-flavor tests of the recently introduced lattice definition and algorithmic implementation of the $\beta$-function defined
on the gradient flow of the gauge field over infinite Euclidean space-time in the continuum.
Originally we introduced this new algorithm to match finite-volume step $\beta$-functions in massless near-conformal gauge theories with the infinite-volume $\beta$-function reached in the chiral limit from small fermion mass deformations of spontaneous chiral symmetry 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 in infinite volume with gradient flow based three-loop perturbation theory, serving as a first pilot study toward the long-term goal of developing
alternate approach to the determination of the strong coupling $\alpha_s$ at the Z-boson pole in QCD. Without reporting here, our tests of this long-term goal continue in QCD with three massless fermion flavors and in the SU(3) Yang-Mills limit of quenched QCD.