The equality characterization for the N-volume quantum Fisher information conjecture

Let $N\in{\mathbb N}$, $\rho\in{\cal D}_n^1$, and $A_1,\ldots,A_N\in M_{n,sa}$. Let $A_j{}_0$ denote the centered observable associated with $A_j$, and let $f\in{\cal F}_{op}^{\,r}$ be a regular symmetric normalized operator monotone function.

Equality characterization conjecture. Equality in the N-volume inequality holds if and only if $A_1{}_0,\ldots,A_N{}_0$ are linearly dependent.

The claim gives the proposed equality case for the preceding volume inequality. The source does not state that this characterization has been proved or disproved.