KanekoShigeki reality criterion for val values

Prove that for every hyperbolic element \gamma\in PSL_2(\mathbb{Z}), the value \operatorname{val}(\gamma) is real if and only if \gamma is conjugate in PSL_2(\mathbb{Z}) to either \gamma^\ast or (\gamma^\ast)^{-1}, where \gamma^\ast is obtained by changing the signs of the off-diagonal entries of a matrix representative of \gamma.

Background

The paper records symmetries of the val function: \operatorname{val}(\gamma^{-1})=\operatorname{val}(\gamma) and \operatorname{val}(\gamma^\ast)=\overline{\operatorname{val}(\gamma)}. These symmetries show that conjugacy to \gamma^\ast or (\gamma^\ast)^{-1} is sufficient for reality of the val value.

Kaneko and Shigeki conjectured that this condition is also necessary. The paper verifies the equivalence between the matrix formulation and the original formulation in terms of real quadratic irrationalities, but explicitly assumes rather than proves the conjecture when deriving the density-zero result for real values.