Conjecture on enumerating permutations avoiding classical and arrow patterns

Let $a_n(\sigma,\tau)$ denote the number of permutations in the relevant class avoiding the classical pattern $\sigma$ and arrow pattern $\tau$. For $n\geq 2$, let $F_n$ be the $n$-th Fibonacci number and $M_n$ the $n$-th Motzkin number.

Enumeration conjecture.

$$a_n(123,(12;1\to3))=2^n-n,$$ $$a_n(321,(12;1\to3))=F_{2n-1},$$

and

$$a_n(321,(12;1\to2))=M_n.$$

These conjectured formulas concern cases combining classical pattern avoidance with arrow pattern avoidance. The surrounding discussion identifies these as examples of sequences that appear in unexplored cases; the paper does not provide a proof or resolution.