An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients

We denote by $M_Z^{\downarrow}(n)$ (resp., $K_Z^{\downarrow}(n)$) the smallest value of $a_0$ that can occur in a non-negative trigonometric polynomial
$$ \sum_{k=0}^n a_k\cos(kx) $$
with non-negative integer coefficients $a_1\geqslant a_2\geqslant\dots\geqslant a_n$ such that $a_n\geqslant 1$ (resp., $\sum_{k=1}^n a_k=n$). We prove that for all natural numbers $n\geqslant 3$
$$ \dfrac{\ln^2 n}{\ln\ln n}\ll K_Z^\downarrow(n)\ll M_Z^\downarrow(n)\ll(\ln n)^3. $$