When two Gaussian random vectors are jointly Gaussian

Assume that $X$ and $Y$ are two Gaussian random vectors with arbitrary finite dimensions (not necessarily the same). It is well known that when both are jointly Gaussian the conditionals $X\mid Y$ and $Y\mid X$ are also Gaussian.

Now let me assume that $X$ and $Y$ are two Gaussian random vectors with arbitrary finite dimensions (not necessarily the same). Furthermore, let me assume that the conditionals $X\mid Y$ and $Y\mid X$ are both Gaussians. Is it possible to prove that $X$ and $Y$ has to be jointly Gaussian?

That is, we need to assert that when $X$ and $Y$ are two Gaussian random vectors with nonsingular covariances, the conditionals are Gaussian if and only if $X$ and $Y$ are jointly Gaussian.

The available proofs (Wikipedia) goes in one direction only.

• If $X,Y$ are jointly Gaussian, then the conditional distribution of $Y$ given the event $X=x$ has the same variance for every value of $x.$ And if the condition distribution of $Y$ given $X=x$ has the same variance for every value of $x$ and the expected value is an affine function of $X$, then one can show $X,Y$ are jointly Gaussian. So can we show that if $Y\mid X=x$ is Gaussian for every $x$ and $X\mid Y=y$ for every $Y,$ then the variance of $Y$ given $X=x$ does not depend on $x\text{?} \qquad$ – Michael Hardy Jul 3 '17 at 17:45
• $\ldots\,$or equivalently and perhaps more suggestively (on how to proceed), can one show that non-constant conditional variance of $X$ given $Y=y,$ as a function of $y,$ prevents the conditional distribution of $Y$ given $X=x$ from being Gaussian? $\qquad$ – Michael Hardy Jul 3 '17 at 18:00