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Benjamin, in fact, almost five years later ..., I can show that any Finsler metric on $SU(N)$ $(N > 2)$ whose geodesics coincide as point sets with those of the (unique up to multiples) bi-invariant Riemannian metric is just a multiple of the bi-invariant Riemannian metric. The same is true for any compact, simple group that has rank greater than one as symmetric space.