Bounded-constant conjecture for small values of bounded-spectrum functions

Let $f$ be a trigonometric polynomial with $n+1$ terms, and let $H_f$ denote the maximum modulus of its nonconstant coefficients. Write $J_f(u)$ for the measure of the set where $|f|<u$, and let $C_n$ be the constants in the previously established estimate

$$J_f(u)\leq C_n H_f^{1/n}u^{1/n}.$$

Bounded-constant conjecture. In the estimate above, $C_n$ can be replaced by a bounded sequence.

The claim would substantially improve the quantitative small-value estimates proved in the paper, whose constants are currently obtained recursively and are only known from computation to have a bounded normalized limit. Its resolution is not supplied here.