Tight bounds for $L_1$ norm of Hermite polynomial: $\int_{-\infty}^\infty |\operatorname{He}_n(x)| \frac{1}{\sqrt{2 \pi}} \exp(-\frac{x^2}{2} ) dx$

Question

Is there a closed-form expression (or tight upper bound) for the integral $$C^{(1)}_n =\int_{-\infty}^\infty |\operatorname{He}_n(x)| \frac{1}{\sqrt{2 \pi}} \exp\left(-\frac{x^2}{2} \right) \mathrm dx$$ Here $\operatorname{He}_n$ is a Hermite polynomial (probabilists' version).

Motivation: we know that \begin{align} C_n^{(2)}=\sqrt{\int_{-\infty}^\infty |\operatorname{He}_n( x)|^2 \frac{1}{\sqrt{2 \pi}}\exp\left(-\frac{x^2}{2} \right) \mathrm dx }= \sqrt{ n!} \end{align}

I am curious in how different is $C^{(1)}_n$ from $C^{(2)}_n$. Using Jensen's inequality, it is not difficult to show that \begin{align} \frac{C_n^{(1)}}{C_n^{(2)}} \le 1 \end{align}

I am curious if there is a more refined bounds, for example, the one that would exactly characterize what \begin{align} \lim_{n \to \infty} \frac{C_n^{(1)}}{C_n^{(2)}} =?? \end{align}