Minimum number of additive tuples in groups of prime order

For a prime number $p$ and a sequence of integers $a_0,dots,a_kin {0,1,dots,p}$, let $s(a_0,dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,dots,x_k)in A_0timesdotstimes A_k$ with $x_0=x_1+dots + x_k$, over subsets $A_0,dots,A_ksubseteqmathbb{Z}_p$ of sizes $a_0,dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=dots=a_k=:a$ and $A_0=dots=A_k$, provided $k$ is not equal 1 modulo $p$. By applying basic Fourier analysis, we show for Bajnoks problem that if $pge 13$ and $ain{3,dots,p-3}$ are fixed while $kequiv 1pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.