Conjecture on pattern-avoiding ascent sequences under the maximum bijection

Let $\operatorname{A}_0(p)$ denote the set of ascent sequences avoiding the pattern $p$, and let $\operatorname{{\bf Sym}}(p_1,p_2)$ denote the set of permutations avoiding both patterns $p_1$ and $p_2$. The map $\mathrm{hat}_{\max}$ is a map from ascent sequences to permutations.

Pattern-avoidance conjecture. The map $\mathrm{hat}_{\max}$ restricts to the following bijections:

$$\operatorname{A}_0(123)\longrightarrow\operatorname{{\bf Sym}}(123,213),$$ $$\operatorname{A}_0(112)\longrightarrow\operatorname{{\bf Sym}}(213,312),$$ $$\operatorname{A}_0(121)\longrightarrow\operatorname{{\bf Sym}}(213,231),$$ $$\operatorname{A}_0(213)\longrightarrow\operatorname{{\bf Sym}}(213,45123).$$

These bijections would identify four classes of pattern-avoiding ascent sequences with corresponding classes of pattern-avoiding permutations. The paper presents them as conjectures, and no resolution is supplied here.