If we change the order of reflection by exchanging the values for the proximal points, r̄₁ and r̄₂, and recomputing we get a second least action solution.
A check shows the reflection points are on the mirrors and the direction cosine requirements for reflection are met. The two solutions are nearly identical except for the relative positions of the reflection points. The total of the path lengths is the same for both problems.
*projections on left should read ρ₁·B₁
Here's the plot in the y,z-plane.
*edit (Jul 2): typo
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