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We use a diffusion Monte Carlo method to solve the many-body Schrödinger equation describing fully-heavy tetraquark systems. This approach allows to reduce the uncertainty of the numerical calculation at the percent level, accounts for multi-particle correlations in the physical observables, and avoids the usual quark-clustering assumed in other theoretical techniques applied to the same problem. The interaction between particles was modeled by the most general and accepted potential, i.e. a pairwise interaction including Coulomb, linear-confining and hyperfine spin-spin terms. This means that, in principle, our analysis should provide some rigorous statements about the mass location of the all-heavy tetraquark ground states, which is particularly timely due to the very recent observation made by the LHCb collaboration of some enhancements in the invariant mass spectra of $J/ψ$-pairs. Our main results are: (i) the $cc\bar c\bar c$, $cc\bar b\bar b$ ($bb\bar c\bar c$) and $bb\bar b \bar b$ lowest-lying states are located well above their corresponding meson-meson thresholds; (ii) the $J^{PC}=0^{++}$ $cc\bar c\bar c$ ground state with preferred quark-antiquark pair configurations is compatible with the enhancement(s) observed by the LHCb collaboration; (iii) our results for the $cc\bar c\bar b$ and $bb\bar c\bar b$ sectors seem to indicate that the $0^+$ and $1^+$ ground states are almost degenerate with the $2^+$ located around $100\,\text{MeV}$ above them; (iv) smaller mass splittings for the $cb\bar c\bar b$ system are predicted, with absolute mass values in reasonable agreement with other theoretical works; (v) the $1^{++}$ $cb\bar c\bar b$ tetraquark ground state lies at its lowest $S$-wave meson-meson threshold and it is compatible with a molecular configuration.