The centrosymmetric growth-rate inequality for permutation classes

Let $\mathcal{C}$ be a permutation class. Write $\overline{\operatorname{gr}}(\mathcal{C})$ for its upper growth rate and $\overline{\operatorname{gr}}^{rc}(\mathcal{C})$ for the upper $rc$-growth rate, obtained from the even-size centrosymmetric permutations in $\mathcal{C}$. Centrosymmetric growth-rate conjecture.

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

The conjecture asserts that the growth of even-size centrosymmetric elements cannot exceed the overall growth of the class. The paper proves the reverse inequality in several settings, including sum-closed $rc$-invariant classes and certain geometric grid classes, but the general inequality remains open.