Completeness bound for positive linear recurrence sequences with coefficients $[1,1,0^k,N]$

Let $(H_n)$ be the positive linear recurrence sequence generated by the coefficient list $[1,1,\underbrace{0,\dots,0}_{k},N]$. Let $f_1=1$ and $f_2=2$, with $f_n$ denoting the Fibonacci numbers, and let $\lfloor\cdot\rfloor$ denote the floor function. Completeness bound conjecture. The sequence $(H_n)$ is complete if and only if

$$1\leq N\leq \left\lfloor\frac{f_{k+6}-k-5}{4}\right\rfloor.$$

This conjecture gives the proposed exact range of the final coefficient for completeness in this family; the necessary condition is stated to be provable, while the sufficiency is conjectural.