The sum-closed centrosymmetric growth-rate equality conjecture

Let $\mathcal{C}$ be a sum-closed and $rc$-invariant permutation class. Let $\operatorname{gr}(\mathcal{C})$ denote its growth rate and $\operatorname{gr}^{rc}(\mathcal{C})$ the growth rate of its even-size centrosymmetric elements, when the latter exists. Sum-closed centrosymmetric growth-rate conjecture. The $rc$-growth rate exists and

$$\operatorname{gr}^{rc}(\mathcal{C})=\operatorname{gr}(\mathcal{C}).$$

The paper proves the inequality $\underline{\operatorname{gr}}^{rc}(\mathcal{C})\geq\operatorname{gr}(\mathcal{C})$ for such classes; this conjecture would follow from the general centrosymmetric growth-rate inequality and remains open.