Finite-correspondence conjecture for Künneth and sign projectors

Let $X/k$ be a projective smooth variety, set $d=\dim X$, and let

$$Z_{\mathrm{fin},\mathrm{H}^{*}}^d\subseteq\mathrm{H}^{2d}(X\times_k X)(d)$$

be the $\mathbb{Q}$-subspace spanned by cohomology classes of codimension-$d$ cycles on $X\times_k X$ that are finite in both projections to $X$. Let $\pi^i_X$ be the Künneth projector and let $\pi^+_X$ be the sign projector. Finite-correspondence conjecture. For every projective smooth variety $X/k$ and every $i\in\mathbb{Z}$, the Künneth projector $\pi^i_X$, respectively the projector $\pi^+_X$, belongs to $Z^d_{\mathrm{fin},\mathrm{H}^{*}}$. This asks whether the Künneth and sign conjectures can be proved using only finite correspondences; the source presents it as a question for more general smooth projective varieties.