Monotonicity conjecture for weighted power means

Let $X$ and $Y$ be positive operators, and for $p\geq2$ define the weighted power mean $(tX^p+(1-t)Y^p)^{1/p}$ for $0\leq t\leq1$. Power-mean monotonicity conjecture. The function

$$t\longmapsto \operatorname{Tr}\bigl(Y-(tX^p+(1-t)Y^p)^{1/p}\bigr)^2$$

should be monotonically increasing on $[0,1]$. This extends the result stated in the source for $1\leq p\leq2$; numerical evidence supports the larger range, while the source does not state a resolution.