Let $A=Doplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|t|^p)^{q/p}=|A+tB|_q^q$ for all $tinmathbb R,$ then $B_{12}=0$ where $B_{12}$ is in the $(1,2)$ block of $B$?

$|.|_q$ denotes the Schatten-q norm and $1<p,qneq 2<infty$

I somehow think that the answer is yes but cannot really prove it.