Given a triangulated category $D$, there is a Chern character from the Grothendieck group to the Hochschild homology: $$ch:K_0(D) \to HH_0(D).$$ Consider a pair of projective Calabi-Yau threefolds $X$ and $Y$ which satisfy homological mirror symmetry. This means that there is an equivalence of triangulated categories $$D Fuk(X) \to D^b Coh(Y)$$ This induces isomorphisms $K_0(X)\to K_0(Y)$ and $HH_0(X)\to HH_0(Y)$. In terms of the Hodge decomposition, $$HH_0(DFuk(X)) \simeq H^{0,3}(X)\oplus H^{1,2}(X) \oplus H^{2,1}(X) \oplus H^{3,0}(X)$$ $$HH_0(D^bCoh(Y)) \simeq H^{0,0}(Y) \oplus H^{1,1}(Y) \oplus H^{2,2}(Y) \oplus H^{3,3}(Y).$$ The Chern character for $Y$ is the standard Chern character of coherent sheaves.

Question 1: Is the chern character for $X$ given by taking the (Poincare dual of the) fundamental class of a Lagrangian submanifold?

Question 2: Is the resulting square of Chern characters commutative?

A related question is whether the isomorphism on $HH_0$ respects the direct sum decompositions from Hodge theory. This seems unlikely because the fundamental class of a Lagrangian is defined over $\mathbb R$, so it is invariant under the complex conjugation which swaps the summands of $HH_0(X)$. On the other hand, sky scraper sheaves on $Y$ have Chern character supported on $H^{3,3}(Y)$.

If indeed the isomorphism does NOT respect the Hodge decompositions, then what do they each correspond to on the other side?