Let $\Sigma$ be Radó's Busy Beaver function, and let $\Delta[\Sigma]$ denote the forward difference of $\Sigma$, such that $\Delta[\Sigma] \ (n) = \Sigma(n+1) - \Sigma(n)$ for all $n \in \mathbb{N}$. Define $\Delta^0[\Sigma] = \Sigma$, and $\Delta^{k+1}[\Sigma] = \Delta[\Delta^{k}[\Sigma]]$, for all $k \in \mathbb{N}$.
Definition: Given functions $f,g$ with the common domain $\mathbb{N}$ and codomains that are subsets of $\mathbb{Z}$, $g$ is said to eventually dominate $f$ iff $g(n) > f(n)$ for all sufficiently large $n\in\mathbb{N}$.
Question: Is it the case that for all $k \in \mathbb{N}$, the function $\Delta^k [\Sigma]$ eventually dominates every computable function $f: \mathbb{N} \rightarrow \mathbb{N}$?
(I believe this is the case, but do not know how to prove it. Any pointers or sources on proving this -- or disproving it, if I'm wrong?)
Here's a difference table showing all the known values for these sequences:
n: 0 1 2 3 4 5 ... ------------------------------------------------ Σ: 0 1 4 6 13 ≥4098 ... ΔΣ: 1 3 2 7 ≥4085 ... ΔΔΣ: 2 -1 5 ≥4078 ... ΔΔΔΣ: -3 6 ≥4073 ... ΔΔΔΔΣ: 9 ≥4067 ... ΔΔΔΔΔΣ: ≥4058 ...
Known sources: "A note on Busy Beavers and other creatures" (1996), by Ben-Amram, Julstrom, and Zwick, contains the conjecture that for any computable function $f$, there exists a constant $N_f$ such that $\forall n \gt N_f, \Sigma(n+1) > f(\Sigma(n))$. In support of this they prove the weaker result that for any computable function $f$, $\Sigma(n+1) > f(\Sigma(n))$ for infinitely many values of $n$. (Considering $f(x) = 2x$, for example, the conjecture implies that $\Delta[\Sigma]$ eventually dominates every computable function.)
Motivation: In the "Busy Beaver game" class of Turing machines, which eventually halt after starting with a blank tape, the "score" of a machine is the number of $1$s remaining on the tape after halting. A variant of Kolmogorov complexity $C: \mathbb{N} \rightarrow \mathbb{N}$ defines $C(n)$ to be the least $k$ such that $n$ is the score of some $k$-state machine. (cf "Computability and Logic" by Boolos, Burgess & Jeffrey, 2007, p229.) I believe I can show that if the first difference $\Delta[\Sigma]$ eventually dominates every computable function, then this $C$ is not monotonically nondecreasing, i.e., there exist $m \lt n$ such that $C(m) \gt C(n)$.
NB: Due to receiving no satisfactory answer here, I've posted the simplest case of this question on cstheory.SE.