Let $BB:\mathbb Z_{\ge1}\to\mathbb Z$ be the Busy Beaver sequence, usually called the Busy Beaver function, as defined in terms of Turing machines in Section 1.3 of this text of Aaronson and Yedidia. We have in particular $BB(1)=1$, $BB(2)=6$, $BB(3)=21$, $BB(4)=107$.
The derivative of a function $f:\mathbb Z_{\ge1}\to\mathbb Z$ is the function $f':\mathbb Z_{\ge1}\to\mathbb Z$ defined by $$ f'(n):=f(n+1)-f(n), $$ and, for $k\in\mathbb N$, the function $f^{(k)}:\mathbb Z_{\ge1}\to\mathbb Z$ is defined by $f^{(k)}:=f$ if $k=0$ and $f^{(k)}:=(f^{(k-1)})'$ if $k\ge1$. (Here $\mathbb Z_{\ge1}$ is the set of positive integers.)
Let $k$ and $n$ be positive integers. Is the integer $BB^{(k)}(n)$ positive?