Minimal sofic shift on a group that is not finitely-generated

We prove that there exists a group which is not finitely generated, but admits a minimal sofic shift. This answers a question of Doucha, Melleray and Tsankov. The group is of the form $(F_4\times F_2)⋊F_{\mathrm{\infty }}$. The construction itself is based on simulation theory and properties of Thompson’s~$V$.