On the universal $\mathrm{CH}_0$ group of cubic threefolds in positive characteristic

We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $\theta^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on surfaces defined over finite fields predicts, via a theorem of Schoen, that every smooth cubic hypersurface of dimension $3$ over the algebraic closure of a finite field of characteristic $>2$ admits a Chow-theoretic decomposition of the diagonal.