Uniqueness of spanning tree on a grid. When I was at the Graduate Student Combinatorics Conference earlier this month, someone introduced me to a puzzle game called Noodles!.
The game starts with a collection of "pipes" on a grid (centered on each vertex), clicking on a piece rotates it $90^\circ$, and a piece can be rotated any number of times. The goal is to turn the final configuration of pipes into a spanning tree (of the grid graph), as shown in the screenshots below.
Example

Question
We left the conference with an unsolved question:
Are solutions to this puzzle always unique? Or is it possible to come up with a starting configuration (on any size grid) that has multiple trees as solutions?
(The prevailing guess is that solutions are unique, but nobody could manage to prove it.)

No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
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