Archer–Geary conjecture on chain-avoiding permutations

Let $C_n(231,1432: 231)$ be the set of permutations in $S_n$ that avoid the chain $(231,1432: 231)$, and let $c_n(231,1432: 231)$ denote its cardinality. Thus, the permutation avoids both $231$ and $1432$, while its square avoids $231$. Let $L_{n+1}$ denote the $(n+1)$-th Lucas number. Archer–Geary's conjecture. For every positive integer $n$,

$$c_n(231,1432: 231)=L_{n+1}-\left\lceil \frac{n}{2}\right\rceil-1.$$

This conjecture concerns the enumeration of permutations satisfying simultaneous pattern-avoidance conditions on a permutation and its square. The source attributes it to Archer and Geary; no resolution is supplied here.