Equality of rank-dimension and coarse dimension for quantifier-free types

Let $(F,\operatorname{Frob})\in\mathcal{S}$. For any countable set $A\subseteq F$ and complete quantifier-free $\mathcal{L}_\sigma$-type $p(x)\in S^{qf}_n(A)$, the two dimensions are defined by the rank-dimension $\operatorname{dim}_{rk}$ and the coarse dimension $\pmb{\delta}_F$. Equality conjecture. The following holds

$$\pmb{\delta}_F(p(x))=\operatorname{dim}_{rk}(p(x)).$$

The conjecture asks whether the two integer-valued additive dimensions coincide on quantifier-free types. The inequality $\pmb{\delta}_F\leq\operatorname{dim}_{rk}$ is established in the paper, while the reverse inequality is not known even in this quantifier-free setting; a positive answer would allow the equivalence to be extended to existential types.