Monotonicity conjecture for lower growth bounds of valence sequences

Let $\sigma_1$ and $\sigma_2$ be valence sequences with $\sigma_1<\sigma_2$, and let $\underline{\lambda}_\sigma$ denote the lower growth-rate bound among face-homogeneous tessellations with valence sequence $\sigma$. Lower-growth monotonicity conjecture.

$$\underline{\lambda}_{\sigma_1}\leq\underline{\lambda}_{\sigma_2}.$$

The authors describe this as a possibly easier conjecture related to the stronger comparison involving the upper bound for $\sigma_1$. The source does not state whether it has been resolved.