Fibonacci enumeration conjecture for simple permutations sortable by two parallel pop stacks

For each $n\geq 0$, let $a_n$ be the number of simple permutations of size $n$ that can be sorted by a machine consisting of two pop stacks in parallel, with entries allowed to bypass the pop stacks. Let $F_n$ denote the $n$-th Fibonacci number.

Fibonacci enumeration conjecture. The sequence $(a_n)$ satisfies

$$a_0=a_1=1,\qquad a_2=2,$$ $$a_n=F_{2n-5}-1\quad\text{if $n\geq 3$ is odd},$$ $$a_n=F_{2n-5}\quad\text{if $n>3$ is even}.$$

This conjecture proposes a closed form for the enumeration of simple sortable permutations in this two-pop-stack model; the source presents it as a conjecture and gives no evidence of a resolution.