The Clergyman's conjecture for linear-form anti-recurrence sequences

Let $cmathbf a=(a_1,\cdots,a_k)$ be a positive integral vector of dimension $k>1$, and let $f(cmathbf x)=cmathbf acdotcmathbf x$ be its linear form. Let $A_n$ be the anti-recurrence sequence for $f$. Let $ctau$ be the trace of the linear form and set $ckappa=kctau+1$. The Clergyman's conjecture. The sequence $A_n-ckappa n$ is $ctau$-automatic. The conjecture is proved in the paper under a restriction on $cmathbf a$; it was previously known for $cmathbf a=(1,1)$, $(1,1,1)$, and $(1,1,1,1)$, so the unrestricted statement remains open.