Al-Rashed–Zegarliński conjecture on monotonicity of the Schatten quadratic form

Let $H$ be a complex Hilbert space and let $B(H)$ denote the linear operators on $H$. For $1<p<\infty$, define the Schatten norm by $\|T\|_p=(\operatorname{Tr}|T|^p)^{1/p}$. For $A,B,D\in B(H)$, define the quadratic form

$$\langle A,B\rangle_D:=\left.\partial_s\partial_t\right|_{s=t=0}\|D+sA+tB\|_p^2.$$

Al-Rashed–Zegarliński's quadratic-form conjecture. For every completely positive trace-preserving map $\phi$ on $B(H)$ and all $A,B,D\in B(H)$,

$$\langle\phi(A),\phi(A)\rangle_{\phi(D)}\leq\langle A,A\rangle_D.$$

The source introduces this as a special case of a conjecture concerning monotonicity of a noncommutative Luxembourg norm. The supplied text does not state whether this conjecture is proved or disproved, so its status remains open here.