Archer et al.'s enumeration conjectures for cyclic permutation pattern avoidance

Let $\pi$ be a cyclic permutation in $S_n$, written in one-line notation as $\pi_1\pi_2\cdots\pi_n$ and in cycle form as $(c_1,c_2,\ldots,c_n)$. The standard cycle form is the cyclic rotation beginning with $1$. Let $\mathcal{A}_n(\sigma_1,\ldots,\sigma_k;\tau)$ be the set of cyclic permutations in $S_n$ that avoid each pattern in $\{\sigma_1,\ldots,\sigma_k\}$ in one-line form and avoid $\tau$ in standard cycle form, and write

$$a_n(\sigma_1,\ldots,\sigma_k;\tau)=|\mathcal{A}_n(\sigma_1,\ldots,\sigma_k;\tau)|.$$

Archer et al.'s conjectures. For the indicated values of $n$, the following enumerations hold:

$$a_n(3421,4321;213)=F_{2n-3}\quad\text{for every integer }n\geq2,$$

where $F_n$ is the $n$-th Fibonacci number, and

$$a_n(1324,1423;213)=\binom{n}{3}+1\quad\text{for every positive integer }n\geq3.$$

These conjectures concern simultaneous pattern avoidance in the one-line and standard cycle forms of cyclic permutations. The source presents them as proposed further directions for research, and no resolution is supplied here.