Domination of prefix sums for surjective binary cellular automata

Let $F$ be a surjective binary cellular automaton with radius $r$, and let $N^1_1(F,r,k)$ denote the corresponding symbol-frequency quantity; write $\mathrm{id}$ for the identity cellular automaton. For $0\leq n\leq r+1$, Domination of prefix sums.

$$\sum_{k=0}^n N^1_1(F,r,k)\geq\sum_{k=0}^n N^1_1(\mathrm{id},r,k).$$

This conjecture would close, in the binary case, the gap between the stated one-step and high-domination results. The source motivates it by computer searches, but gives no proof or resolution.