Stretched Billboard particles only render when in motion

Check out this screencast to see my problem in action: https://www.youtube.com/watch?v=f_mJFZOE4Is&feature=youtu.be&hd=1

When the particles are set to Billboard at the beginning, they act as I intended: They fade in, stop moving, wait a few seconds, then fade out.

As soon as I switch it to Stretched Billboard, they start disappearing as soon as they've stopped moving. In the screencast, I help to illustrate this by making them fade out earlier (before they totally stop moving).

Why does Stretched Billboard only render when in motion, while Billboard does not? I'm not using the SpeedScale property (it is 0), and when I change it to other values, including -1 or 1, the problem persists. Is there any way around this?