Generalized self-concordance of the log trace-inverse function

Let $p>0$, let $X$ range over a subset of the positive-definite cone, and define

$$k(X)=\log\left(\operatorname{Tr}\left(X^{-p}\right)\right).$$

Log trace-inverse self-concordance conjecture. The function $k(X)$ is $(\nu,M)$-generalized self-concordant for some $\nu\in[2,3]$ and $M>0$ on some subset of the positive-definite cone. The source says that computational experiments strongly suggest this property, while noting that the preceding proof for the nonlogarithmic A-criterion does not translate to the log variant.