Furuta's matrix trace inequality conjecture

Let $M(n,\mathbb{C})$ be the set of $n\times n$ complex matrices, let $M_h(n,\mathbb{C})$ be the set of Hermitian matrices, and let $M_+(n,\mathbb{C})$ be the set of positive semidefinite matrices. For $X,Y\in M_+(n,\mathbb{C})$, $p\in\mathbb{R}$, and $I$ the identity matrix, consider the two trace inequalities below.

Furuta's conjecture. It was asked whether

$$\operatorname{Tr}\left[(I+X+Y+Y^{1/2}XY^{1/2})^p\right]\leq \operatorname{Tr}\left[(I+X+Y+XY)^p\right]$$

for $p\geq 1$, and whether

$$\operatorname{Tr}\left[(I+X+Y+Y^{1/2}XY^{1/2})^p\right]\geq \operatorname{Tr}\left[(I+X+Y+XY)^p\right]$$

for $0\leq p\leq 1$.

The source paper states that its purpose is to answer this question, so the candidate is presented as a conjectural question rather than an asserted theorem. The surrounding excerpt does not establish whether the question was open at the time of the source paper or whether the source resolves it.