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.
...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
[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.