Growth conjecture for simple permutations in the avoidance class

Let $s_n$ denote the number of simple permutations of length $n$ in the class described above.

Growth conjecture. For $n \geq 12$,

$$s_n \geq 9s_{n-2}.$$

If true, this would improve the lower bound on the exponential growth rate of the numbers $c_n$ by providing stronger lower bounds for the number of simple permutations.