The Golomb return conjecture for threshold positions

Let $G$ be Golomb's sequence, let $S(k)=\sum_{n=1}^{k}G(n)$, and let $N_r(n)$ denote the relevant multiplicity for order $r$; define $M(r)=\sup_n N_r(n)$. For each $k$, let $j_k$ be the smallest order $r$ at which $M(r)$ first reaches $k$. Golomb return conjecture. For all $k\ge 4$,

$$j_{k+1}-j_k=G(k)\qquad\text{and}\qquad j_k=S(k-1)+2.$$

This conjecture describes the observed Golomb-sequence structure of the threshold positions $j_k$. A complete proof remains open; the paper reduces it to two boundary conjectures.