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Current and future B-mode polarization data are the most powerful observables to constrain gravitational waves from the early Universe. We set conservative constraints on tensor modes when relaxing the inflationary consistency condition $n_t=-r/8$ between the tensor tilt $n_t$ and the tensor-to-scalar ratio r. By adding a power-law spectrum of tensor perturbations to $Λ$CDM, and parameterizing this tensor contribution by two independent primordial tensor-to-scalar ratios $(r_1,r_2)$ at $k_1 = 0.005$ Mpc$^{-1}$ and $k_2 = 0.02$ Mpc$^{-1}$, Planck and BICEP/Keck Array 2018 data (BK18) lead to constraints $r_{0.005} < 0.030$ and $r_{0.02} < 0.098$ at 95% CL. The corresponding upper bound $r_{0.01} < 0.039$ is by a factor of 2 tighter than the one obtained with Planck 2018 and the older BK15 data. We then study the perspectives for future CMB experiments that will measure both the reionization bump and recombination peak of the B-mode polarization angular power spectrum, such as LiteBIRD. We test the robustness of the results to the choice of the scales for $(r_1,r_2)$ in these future perspectives. Whereas distinguishing $n_t=-r/8$ from exact scale invariance is impossible as expected, we show how radical, theoretically motivated departures from $n_t=-r/8$, which are consistent with the current data, could be distinguished with LiteBIRD. Moreover, LiteBIRD will be able to shrink the allowed parameter space area in the $(r_{0.005},r_{0.02})$ plane to less than one hundredth of the currently allowed area by Planck 2018 and BK18.