The contrast of paper, in your case canson baryta is about 250, may be between 250-300 assuming your canon printer can deliver deep black. Your monitor is probably more in the range of 700-100. So yes dramatic contrast reduction(blacks, less black)
Does a print wrt colors/tints etc, including the colorshifts look sort of the same to the softproof result?
If so then the printerprofile for that paper-printer combination may have a coloshift problem.
Also the baryta paper is in certain areas capable of handling more saturated colors (bigger than aRGB) than your monitor can display (i do not know about your canon printer, but in case of HP B9180 or Epson4900 it certainly is).
How your monitor displays colors that are out of its gamut is unknown, but can be another problem cause. One a Dell 2412 or 2312 it can lead to a exagaration of those colors, making it look not very nice.
If you profile your camera with ColorCheckerPassport, it can lead to relatively high levels of saturation, depending on the objects you photograph, like flowers, foliages (many shades of green, some quite saturated), or man-made stuff like make-up, clothing.
Whether these levels of staturation are correct is not the issue here, but can add to a wrong display on your monitor of those saturated colors.
Personally in cases where i worked with dual-monitor setups, i Always found one monitor to suffer from the other, unless both were identical in model and type.
Lightrom does appear to have problems in dual monitor setups, and yes the softproof function has its quirks.