Alternating-exponential integral conjecture

Let $n$ be a positive integer and let $\alpha_1,\ldots,\alpha_{2n}$ be real numbers satisfying

$$\alpha_1\leq\alpha_2\leq\cdots\leq\alpha_{2n}.$$

Consider the alternating exponential sum $\sum_{j=1}^{2n}(-1)^j e^{i\alpha_jt}$. Alternating-exponential integral conjecture. One has

$$\int_1^{\infty}\frac{\left|\sum_{j=1}^{2n}(-1)^j e^{i\alpha_jt}\right|^2}{t^2}\,dt\geq\int_1^{\infty}\frac{\left|1-e^{i\sum_{j=1}^n(\alpha_{2j}-\alpha_{2j-1})t}\right|^2}{t^2}\,dt.$$

The source states that this formulation is equivalent to the characteristic-function rearrangement conjecture. It is therefore an associated open formulation of the same problem, expressed through weighted $L^2$ norms of non-harmonic trigonometric polynomials.