Monotonicity conjecture for completeness in families of positive linear recurrence sequences

Let $g,k$ be nonnegative integers and $N$ an integer, and consider the positive linear recurrence sequences associated with the coefficient lists

$$[\underbrace{1,\dots,1}_{g},\underbrace{0,\dots,0}_{k},N] \quad\text{and}\quad [\underbrace{1,\dots,1}_{g+1},\underbrace{0,\dots,0}_{k},N].$$

Monotonicity conjecture. If the sequence generated by $[\underbrace{1,\dots,1}_{g},\underbrace{0,\dots,0}_{k},N]$ is complete, then so is the sequence generated by $[\underbrace{1,\dots,1}_{g+1},\underbrace{0,\dots,0}_{k},N]$. The authors motivate this from observed numerical behavior: for fixed $k$, the maximal admissible $N$ increases with $g$ and eventually becomes constant. A general proof remains open.