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. 

> ...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.:)

**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]http://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.