Al-Rashed–Zegarliński monotonicity conjecture below Schatten exponent two

Let $H$ be a complex Hilbert space, let $B(H)$ denote the linear operators on $H$, and let $B(H)_{++}$ denote the invertible positive operators. For $\beta\in\mathbb{R}$, $\alpha\in[-1,0]$ with $\alpha+\beta=-1$, $1\leq p<\infty$, and $P\in B(H)_{++}$, define

$$\Lambda_{\alpha,\beta,p}(P,X):=\|P^{\frac{\alpha}{p}}XP^{\frac{\beta}{p}}\|_p^p=\operatorname{Tr}|P^{\frac{\alpha}{p}}XP^{\frac{\beta}{p}}|^p.$$

Al-Rashed–Zegarliński's conjecture. If $p\in[1,2)$, then for every $X\in B(H)$ and every unital completely positive trace-preserving map $\phi$ on $B(H)$,

$$\Lambda_{\alpha,\beta,p}(\phi(P),\phi(X))\leq\Lambda_{\alpha,\beta,p}(P,X).$$

This proposed extension below $p=2$ is refuted in the paper. Monotonicity would imply joint convexity of $\Lambda_{\alpha,\beta,p}$, but the corresponding scalar function $x^p y^{-1}$ is jointly convex only when $p(p-2)\geq0$, which fails for $p\in[1,2)$.