Homomorphism-preserving bijections of finite graphs are trivial

Let $\mathcal{G}$ be the class of finite simple graphs, and let $\pi:\mathcal{G}\rightarrow\mathcal{G}$ be a bijection. For graphs $G,H\in\mathcal{G}$, write $hom(G\rightarrow H)$ for the number of graph homomorphisms from $G$ to $H$.

Homomorphism cancellation conjecture. If, for all graphs $G,H\in\mathcal{G}$,

$$hom(G\rightarrow H)=hom(\pi(G)\rightarrow\pi(H)),$$

then $\pi$ is the identity map.

This is posed as a generalisation of Lovász's homomorphism cancellation laws in the setting of reconstruction from subgraph posets. The supplied text presents it as a problem, and gives no resolution.