Finite homomorphism basis conjecture for first-order equivalence

Let $k\ge 1$. Fix a set $\Gamma$ of colours and consider graphs $G$ whose colour interpretations satisfy $\operatorname{rg}(\gamma^G)\subseteq\Gamma$. For graphs $G,G'$ and a finite set $\mathcal F_k$ of graphs, say that $G$ and $G'$ are homomorphism-indistinguishable over $\mathcal F_k$ when their homomorphism counts from every graph in $\mathcal F_k$ agree. Finite homomorphism basis conjecture. There is a finite set

$$\mathcal F_k\subseteq\mathcal{TD}_k$$

such that, for all graphs $G,G'$, if $G$ and $G'$ are homomorphism-indistinguishable over $\mathcal F_k$, then $G$ and $G'$ satisfy the same $\textsf{FO}_k$-sentences. This asks whether bounded-quantifier-rank first-order equivalence can be captured by finitely many homomorphism counts from graphs of tree depth at most $k$; the source presents it as a tempting conjecture, and no resolution is supplied here.