The finiteness conjecture holds in SL(2,Z>=0)^2
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Let A,B be matrices in SL(2,R) having trace greater than or equal to 2. Assume the pair A,B is coherently oriented, that is, can be conjugated to a pair having nonnegative entries. Assume also that either A,B^(-1) is coherently oriented as well, or A,B have integer entries. Then the Lagarias-Wang finiteness conjecture holds for the set A,B, with optimal product in A,B,AB,A^2B,AB^2. In particular, it holds for every matrix pair in SL(2,Z>=0).
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