The eventual quasi-polynomial conjecture for the parametric Frobenius problem

Let $a_i:bbZ_+crightarrowbbZ_+$, $1cle icle n$, be eventually positive functions in $cmathrm{EQP}$, where $F(a_1,\cdots,a_n)$ denotes the largest element of the group $cmathbb{Z}a_1+\cdots+cmathbb{Z}a_n$ not in the semigroup generated by $a_1,\cdots,a_n$. A function is in $cmathrm{EQP}$ if it eventually agrees with a quasi-polynomial. The parametric Frobenius conjecture. Then

$$Fcbig(a_1(t),\cdots,a_n(t)cbig)cincmathrm{EQP}.$$

The conjecture predicts eventual quasi-polynomial behavior for Frobenius numbers whose generators vary as eventual quasi-polynomial functions. The paper proves this in the special cases where the generators are linear functions or where $ncle 3$, while the general case remains open.