New aspects of strong Hom-schemes and strong G-schemes

Let $\mathfrak{P}$ be the class of all finite posets. For $P, Q \in \mathfrak{P}$, let ${\cal H}(P,Q)$ and ${\cal S}(P,Q)$ be the sets of order homomorphisms and of strict order homomorphisms from $P$ to $Q$, respectively. From an earlier investigation we know, that for $R, S \in \mathfrak{P}$, the relation "$\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for all $P \in \mathfrak{P}$" implies "$\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for all $P \in \mathfrak{P}$". Now we ask for the inverse: For which posets $R, S \in \mathfrak{P}$ does the existence of a poset $P \in \mathfrak{P}$ with $\# {\cal S}(P,R) > \# {\cal S}(P,S)$ imply the existence of a poset $Q \in \mathfrak{P}$ with $\# {\cal H}(Q,R) > \# {\cal H}(Q,S)$? We prove that this is true for posets $P, R$, and $S$ belonging to certain classes. In particular, the implication holds for all flat posets $R$ and $S$.