Nonexistence of finitely generated standard elementary extensions of free lattices

Let $\mathbf F_n$ be the free lattice on $n$ generators. An elementary extension $\mathbf K\succ\mathbf F_n$ is standard if the associated map $h_{\mathbf K}$ has range in $\mathbf F_n$. Nonexistence conjecture. There is no finitely generated standard elementary extension of $\mathbf F_n$. This conjecture concerns the structure of models of $\operatorname{Th}(\mathbf F_n)$ and would provide a positive result under the standardness hypothesis, in contrast with the general complexity of these models. The source does not state whether the conjecture is resolved.