The morphic conjecture for even and odd morphic cohomology

Let $X$ be a smooth projective variety of dimension $m$, let ${\mathbb F}$ be the coefficient field used in the source, and let $\widetilde{L}^tH^k(X;{\mathbb F})$ be the image of morphic cohomology in singular cohomology. Define

$$EH^a(X;{\mathbb F})=\widetilde{L}^aH^0(X;{\mathbb F})\oplus\widetilde{L}^{a+1}H^2(X;{\mathbb F})\oplus\cdots\oplus\widetilde{L}^{a+m}H^{2m}(X;{\mathbb F}),$$ $$OH^b(X;{\mathbb F})=\widetilde{L}^bH^1(X;{\mathbb F})\oplus\widetilde{L}^{b+1}H^3(X;{\mathbb F})\oplus\cdots\oplus\widetilde{L}^{b+m-1}H^{2m-1}(X;{\mathbb F}),$$ $$LH^{a,b}(X;{\mathbb F})=EH^a(X;{\mathbb F})\oplus OH^b(X;{\mathbb F}).$$

Let $\mathcal L$ be the restriction of the Lefschetz operator $L$ to these spaces, and let $\lambda$ be its adjoint with respect to the Hodge inner product. Let $\Lambda$ be the adjoint of $L$ on cohomology. Morphic conjecture. On each of $EH^a(X;{\mathbb F})$, $OH^b(X;{\mathbb F})$, and $LH^{a,b}(X;{\mathbb F})$, the operator $\lambda$ is the restriction of $\Lambda$. The source states that these conjectures hold for abelian varieties, complete intersections, and Grassmannians, and proves that Grothendieck standard conjecture B implies them; no general resolution is given.