Divisibility conjecture for one Hermitian-form equation

Prove that the number of solutions of the monic homogeneous Hermitian-form diagonal equation X_1^{q+1}+\cdots+X_{k+1}^{q+1}=0 over (F_{q^4})^{k+1} satisfies N_1(k+1,q)\equiv 0\pmod{q^{2k-1}} for every k\in\mathbb{N}.

Background

The paper derives a closed formula for N_1(s,q), the number of solutions in (F_{q4})s of the equation X_1{q+1}+\cdots+X_s{q+1}=0, and proves the weaker divisibility statement N_1(s,q)\equiv 0\pmod q for every s\ge 2.

Examining the explicit formulas for small values of s, the authors observe the pattern N_1(k,q)=q{2k-3}P_k(q) for a polynomial P_k(q)\in\mathbb{Z}[q]. They formulate the stronger divisibility assertion below as a conjecture, which would substantially improve Proposition 6.1.

References

So, we conjecture that $$ N_1(k+1,q) \equiv 0 \pmod{q{2k-1}$$ for any $k \in N$, thus improving a lot the assertion in Proposition \ref{prop: N_1xs}.

— On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases  (2608.19507 - Podestá et al., 19 Aug 2026) in Remark following Corollary 6.2, Section 6, “The Hermitian-form diagonal equation”