Suppose that $\mathbb{A},\mathbb{B}$ are finitely complete categories and $F:\mathbb{A}\to\mathbb{B}$ is a functor which reflects finite limits. Does $F$ reflect finitely generated limits?

(Here "finite limits" means limits over finite index categories, and "finitely generated limits" means limits over finitely generated index categories. This terminology is from "Handbook of categorical algebra vol. 1" by Borceux.)

This is basically a question about Proposition 2.9.8 from Borceux's "Handbook of categorical algebra vol. 1". I agree with Borceux about the part of the proposition dealing with the preservation of limits: just construct the limit as an equalizer of two arrows between finite products and observe that $F$ maps the whole construction into $\mathbb{B}$ producing a limit of a composition. However, unfortunately I cannot see how does this help with the reflection of limits.