Derivative of Adjugate Matrix, $\frac{d^i}{dx^i}\operatorname {Adj}(x \mathbf I - \mathbf A)$

Question

Let $\mathbf A$ denote an $n \times n$ matrix with $r=\operatorname{rank}\mathbf A$ and define $$ \mathbf B_i(x) \equiv \frac{d^i}{dx^i}\operatorname {Adj}(x \mathbf I - \mathbf A). $$ Conjecture:

If $n - r \ge 2$, then $$ \mathbf B_i(0)= \mathbf 0, \quad \text{for all } i \in \{0,...,n-r-2\}. $$