Nick, I would bet you could hear a difference the first time you power up those Studios with either one. The 2807 and the 3806 are a big step up from the 1705. The question is, once you get to the 3806 level, whether you would hear a difference if you add a 2 X 200 (minimum) wpc amp to power the front speakers.
As you already know, many people reported big, huge, day and night type of differences while others (me included) reported only little or subtle difference especially at low to moderate volume. I should add that the first time I had my amp powering the 87 dB Veritas I thought I heard significantly better sound in terms of clarity. A few months later, out of curiosity, I started moving things around and did some A/B (sighted) listening using different combinations of receiver, amps, preamp, speakers and playing different type of music. I ended up concluding that the differences were subtle. There were a few others around when I did those comparisons and they did not hear much difference either. The significant improvement I initially perceived could have been just Placebo effect. As I stated repeatedly in the past, I do believe some people can hear more "difference".
Back to your question, for peace of mind, go for the 3806. It is more robust for sure. In case you decide not to add any amp, it has the power to do a decent job with the Studio 20, at least in two-channel mode.