Divisibility conjecture for two Hermitian-form equations

Prove that the number of solutions of the monic homogeneous Hermitian-form system X_1^{q+1}+\cdots+X_{k+3}^{q+1}=0 and X_1^{q^3+1}+\cdots+X_{k+3}^{q^3+1}=0 over (F_{q^8})^{k+3} satisfies N_2(k+3,q)\equiv 0\pmod{q^{4k+2}} for every k\in\mathbb{N}.

Background

The paper obtains a closed formula for N_2(s,q), the number of solutions in (F_{q^8})^s of the two-equation Hermitian-form system with exponents q+1 and q^3+1. The proven result gives the weaker divisibility N_2(s,q)\equiv 0\pmod q for every s\ge 2.

The authors note that, beginning with s=4, the observed exponent of the isolated factor q follows the pattern N_2(k+3,q)=q^{4k+2}P_k(q) for some P_k(q)\in\mathbb{Z}[q]. They conjecture that this pattern yields the stronger divisibility property stated below.