Extremal growth-rate conjecture for polymorphic valence sequences

Let $\sigma$ be a polymorphic valence sequence. Let $\mathscr{T}_\sigma$ be the set of isomorphism classes of face-homogeneous tessellations with valence sequence $\sigma$, and define

$$\underline{\lambda}_\sigma=\inf\{\gamma(T):T\in\mathscr{T}_\sigma\},\qquad \overline{\lambda}_\sigma=\sup\{\gamma(T):T\in\mathscr{T}_\sigma\}.$$

Also set

$$\mathscr{L}_\sigma=\{T\in\mathscr{T}_\sigma:\gamma(T)=\underline{\lambda}_\sigma\},\qquad \mathscr{H}_\sigma=\{T\in\mathscr{T}_\sigma:\gamma(T)=\overline{\lambda}_\sigma\}.$$

Extremal growth-rate conjecture. The sets $\mathscr{L}_\sigma$ and $\mathscr{H}_\sigma$ are nonempty. Thus the lower and upper growth-rate bounds are realized by tessellations with valence sequence $\sigma$. The source presents this as an open conjecture and gives no resolution.