The domination lemma for almost Golomb sequences

Let $a_r$ be the order-$r$ almost Golomb sequence, let $N_r(n)$ denote the multiplicity of the value $n$, and define $M(r)=\sup_n N_r(n)$. Domination Lemma. For all $r\ge 5$,

$$M(r)=N_r(r-1).$$

Together with the run-length identity and the Prefix Conjecture, this would identify the global maximum multiplicity with the boundary value at $r-1$. The claim is presented as one of the two boundary conjectures in the conditional reduction, and no proof or resolution is supplied.