  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 arXiv:2004.00754. 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.
