Set-valued characterization of test Morita equivalences

Let $F\colon\mathcal{S}\to\mathcal{T}$ be a morphism of limit sketches. A test Morita equivalence is a morphism of sketches that induces an equivalence

$$F^*\colon \mathsf{Skt}(\mathcal{T},\mathsf{Set})\to\mathsf{Skt}(\mathcal{S},\mathsf{Set})$$

for every test sketch. Set-valued characterization conjecture. The morphism $F$ is a test Morita equivalence if and only if it induces an equivalence

$$F^*\colon \mathsf{Skt}(\mathcal{T},\mathsf{Set})\to\mathsf{Skt}(\mathcal{S},\mathsf{Set}),$$

where $\mathsf{Set}$ is the category of sets with its natural test structure. This conjecture would characterize test Morita equivalences using only models in sets, in the spirit of Gabriel–Ulmer duality; its resolution would clarify whether the full quantification over test sketches can be reduced to the category of sets.