If $Y$ is a normal random variable and $E[Y\mid X]$ is also a normal random variable, does it mean that $X$ is a normal random variable?

Question

Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space. If $Y$ is a (non-degenerate) normal random variable and $E[Y\mid X]$ is also a (non-degenerate) normal random variable, does it mean that $X$ is necessarily a normal random variable ? How would one show that ?

Answer 1

$Y \sim \mathcal N(0,1)$.

$\mathbb E[Y \mid e^Y] = \mathbb E[Y\mid Y] = Y \sim \mathcal N(0,1)$, but $e^Y \not\sim \mathcal N(\mu,\sigma^2)$.