Gear type LSD's (helical, torsen, quaife) do exhibit this yes. The suretrack are viscous, so rely on differences in speed to lock, and therefore won't do this.

Clutch type diffs like ours will do this if there is little/no pre-load (static friction). This does not mean that the diff won't operate correctly when both wheels have (well, at least a little bit of) traction though.

Adding more preload will return the low load/traction loackability of the diff. I guess you could shim the springs or you could also add springs with more force?

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