Years ago, in the early days of baryta papers, I did a little controlled experiment using Andrew Rodney's original printer test image to see how a variety of baryta and non-baryta papers responded to identical inking in my iPF 5000: Canon's "Special #5 media type (the heaviest inking for PK papers), and identical printer driver settings.
I was expecting to see big differences, but when the papers were strewn randomly on a table under good light, hard to tell apart, the differences very slight - mostly dependant on the base tint, with the Polar Satin the whitest. The one outlier was the Harmon, with a slightly yellowish warmer cast than it's base whiteness would predict. But no specific color shifts or other surprises i was expecting to see.
With prints thoroughly dried I scanned each 8.5x11 image on my Epson V700 scanner with identical settings, including white point set to show base paper tint differences. I did a total of nine PK papers, including Ilford, Red River, Hahne., and Harmon brands. Six of these scans are in a Flickr album: R.R. Polar Satin and UltraPro Satin-2; Hahne Fine Art Pearl and Photo Rag Pearl; Ilford Gold Fibre Silk; and Harmon FbAl Gloss:
https://www.flickr.com/photos/8075035@N03/albums/72157624569134778Since then I've been using RR satins for my proofing (currently Palo Duro), and Ilford GFS for display - printed with PS control and the same media type (Special #5), and Canon profile (Canon iPF5100 Photo Paper Plus Semigloss-highest). Except for the rather intangible qualities of depth and "presence", and of course the paper feel, they are quite identical.