The homological and numerical equivalence conjecture for algebraic cycles

Let $X$ be a smooth projective variety and let $C^i_{\sim}(X)$ be the group of codimension-$i$ cycles modulo an equivalence relation $\sim$. Write $C^i_{\sim_{\mathrm{hom}}}(X)$ for cycles modulo homological equivalence and $C^i_{\sim_{\mathrm{num}}}(X)$ for cycles modulo numerical equivalence. Fundamental Conjecture D(X). For every codimension $i$,

$$C^i_{\sim_{\mathrm{hom}}}(X)=C^i_{\sim_{\mathrm{num}}}(X).$$

This conjecture asserts that homological and numerical equivalence coincide for algebraic cycles, a central assertion in the theory of classical motives; the source does not state a resolution.