Comments on conjectures of trace inequalities on a generalized Wigner-Yanase skew information

Furuichi, Yanagi and Kuriyama gave three conjectures of trace inequalities on the Wigner-Yanase-Dyson skew information and a generalized Wigner-Yanase skew information (Furuichi et al. (2009) 1]) and Yanagi found a counterexample showing that two of the three conjectures don't hold (Yanagi (2010) 6]). In this note, we show that the last conjecture does not hold in general. In addition, we show that in the case of 2×2 matrices the conjecture is true.