Differential equations for the exponential generating functions of two vincular avoidance classes

Let $[\Pi]$ be a set of possibly vincular patterns, and define its exponential generating function by

$$E[\Pi]=\sum_{n\ge 0} \#\operatorname{Av}_n[\Pi]\frac{x^n}{n!}.$$

Write $\overline{123}$ and $\overline{213}$ for the two vincular patterns in question, and let $E'=\frac{dE}{dx}$.

Conjectures. For $E=E[\overline{123}]$ and for $E=E[\overline{213}]$, respectively,

$$E'=E^2-E+1,$$

and

$$E'=e^{E-\frac{x^2}{2}}.$$

These differential equations would determine the corresponding avoidance-class enumerations, since their explicit solutions can be found by separation of variables once the equations are proved. The source provides no resolution of either conjecture.