Is the Gaussian Correlation Inequality universal? T. Royen proved the Gaussian correlation inequality in the context of Gamma distributions back in 2014, which was since popularized by Latala and Matlak. The properties of Gaussian integration seem heavily exploited in the original proof: even the reduction from convex sets to cylindrical ones rely on the fact that projections of Gaussian vectors are again Gaussian. Has anyone attempted to prove the result for more general distributions beyond multivariate Gamma, or are there obvious counterexamples? 
 A: I'm not that familiar with this area but there do exist some results for more general probability distributions but more restricted sets.  For example, as explained by Li and Shao, the Gaussian correlation inequality may be reformulated as saying that if $(X_1, \ldots, X_n)$ is a centered, Gaussian random vector, then
$$\mathbb{P}\left(\max_{1\le i\le n} |X_i| \le 1\right) \ge
\mathbb{P}\left(\max_{1\le i\le k} |X_i| \le 1\right)
\mathbb{P}\left(\max_{k+1\le i\le n} |X_i| \le 1\right)$$
for all $1\le k < n$.  The case $k=1$ was proved by Khatri and Šidák (independently), and for this special case, Das Gupta et al. have proved a generalization to elliptically contoured distributions.  Das Gupta et al. also give some counterexamples to overly strong generalizations of the conjecture.
A: Unlike Chow's answer, I do not think the results for elliptically contoured distributions is in the same spirit as GCI because they are controlling the bound of extreme values, which is more like results from U-statistics instead of the generality of GCI.
I think Royden's thinking is basically following Renyi's theorem [5](Or Cramer-Wold if you like) and consequential work in this direction is ongoing using Renyi's divergence applied on convex bodies[4]. 

...even the reduction from convex sets to cylindrical ones rely on the
  fact that projections of Gaussian vectors are again Gaussian.

According to the technique that Royden used, it relies heavily on the fact that the Gamma family is reproducing[1] (OR projection closed, which does not generalize to many other families). The key arguments in his proof, as pointed out by Latala and Matlak[2], is the repeatative use of rectangular sets and the projected images onto these sets.
So I am doubtful that the GCI can be generalize further to other families beyond Gamma. At least I do not believe that these set of techniques can be generalized directly for otherwise Latala and Matlak must have already done.:) There is also another discussion about the application of GCI [3].
Reference 
[1]Teicher, Henry. "On the convolution of distributions." The Annals of Mathematical Statistics (1954): 775-778. https://projecteuclid.org/euclid.aoms/1177728664
[2]Latała, Rafał, and Dariusz Matlak. "Royen's proof of the Gaussian correlation inequality." arXiv preprint arXiv:1512.08776 (2015). https://arxiv.org/abs/1512.08776
[3]https://stats.stackexchange.com/questions/270639/consequences-of-the-gaussian-correlation-inequality-for-computing-joint-confiden
[4]Kumar, M. A., & Sason, I. (2016). Projection Theorems for the Rényi Divergence on $\ alpha $-Convex Sets. IEEE Transactions on Information Theory, 62(9), 4924-4935.
[5]Renyi, Alfréd. "On projections of probability distributions." Acta Mathematica Hungarica 3.3 (1952): 131-142.
