The Clergyman's Conjecture for anti-k-naccis

Let $X_n$ be the sequence of anti-$k$-naccis, defined as sums of $k$ consecutive missing numbers. A sequence is $k$-automatic when its base-$k$ value sequence can be generated by a finite automaton. The Clergyman's Conjecture. The difference

$$X_n-(k^2+1)n$$

is $k$-automatic. This conjecture proposes a uniform automatic description for all anti-$k$-naccis; the paper presents it as an evident pattern but does not establish it in the supplied text.