However, they are not asking the same question, and I think that unless the answers at the supposed duplicate are altered to respond to this question, they don't actually answer it.
The question is asking about a trivariate where all of the bivariate margins are bivariate normal. The question it's marked as a duplicate of is asking about a bivariate where the univariate margins are normal.
An answer to the question "Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?" does not of itself necessarily imply the same answer to the question about bivariate margins of a trivariate distribution. It's a rather subtler question, and the counterexample somewhat more complex than the counterexamples for the supposed duplicate.
While the answers could be extended to answer this question, as far as I can see they currently don't.
When it first came up for closure (a couple of weeks ago), I responded (via a comment) pointing out the difference. I've only just noticed that it was voted for closure in spite of the comment.
I've just nominated to reopen, but apparently my original comment was insufficient, so I raise it here, in the hope that we can resolve this question.