Automaticity of anti-k-nacci deviations

Prove that for every integer k, if X_n denotes the sequence obtained by summing k consecutive missing numbers in the corresponding greedy mex construction, then the difference X_n-(k^2+1)n is k-automatic.

Background

The paper generalizes anti-Fibonacci, anti-Tribonacci, and anti-Teranacci constructions by defining anti-k-naccis as sums of k consecutive missing numbers. For the cases explicitly examined, the authors identify automatic remainder sequences and verify the defining complementary and additive properties computationally.

The stated conjecture proposes a uniform automaticity law for the deviation of the anti-k-nacci sequence from the linear term (k^2+1)n. No proof of this general statement is supplied.