Reality is it doesn't make much difference.
For a static situation, the tires are massively overdesigned, so they don't need help.
Two possible issues with leaving them down:
- If the tires go flat, the stabilizers may become overloaded. Most of the simple scissor style stabilizers are not designed to life the trailer. Also the frame may not be designed to be lifted from the corners.
- When you go to collect the trailer, don't forget to retract the stabilizers.
Not major concerns but something to think about.