Source?
From personal experience - I have yet to see a cyclist "taking due care" when they run the stop sign by where I live. They just run it full speed downhill and cars have to dodge them.
What especially confuses me in this situation is that if the cyclist runs into a car turning left after the car properly stopped and then started moving without see the cyclist speeding downhill (uphill from the car and usually beyond the view due to the said hill), the cyclist is the one with the most to lose. They do it anyways.