There are more factors at play here but the tldr is that, other than wheat, rye flour contains a very high level of amylase, an enzyme that breaks starches into sugars. Amylase prevents bread from rising, and rye contains so much of it that it stays active even with the high temperatures of baking. Add that to the fact that rye has very little gluten to hold up the structure of the bread…
That‘s where sourdough comes into play. The acids produced during a slow fermentation slow down the reaction of amylase, giving you a lighter bread. For a good rye bread, I ferment for over 24h in three steps.
Interesting you say that though, because I find three days about the limit, starts to really over-ferment, becoming really 'pockety', even at refrigeration temperatures. I suppose if I did it all in there as you imply, no fermentation time at room temp. at all, that might be better?
How active is your sourdough starter? Commercial yeast by contrast tends to be very active, so timings are shorter.
That said, I think they cultivate newer rye varieties to be less rich with enzymes. I presume that is to meet the requirements of industrial baking.
(Though you shouldn't use that kind of yeast for breadmaking, if you want it to taste good.. At least get some fresh yeast. (Which not quite coincidentally is also the 'normal' kind of yeast in eg Germany.) Better yet, make a rye sourdough starter.)
The problem is that rye doesn't have enough gluten. So you need to get the structure of your bread from the starches, and they only do the Right Thing when it's sour enough. See https://news.ycombinator.com/item?id=36504484 for a slightly more in-depth description.