Bourgain's conjecture for trigonometric polynomials on the Bohr compactification of $\mathbb R$

Let $\mathcal P(\mathbb R)$ be the space of trigonometric polynomials on $\mathbb R$, and let $(\omega_j)_{j\in\mathbb N}$ be an increasing sequence of real numbers. Define

$$J_{\omega,n}=\left\{P\in\mathcal P(\mathbb R):P(t)=\sum_{k=0}^{n}a_ke^{i\omega_kt},\ a_0=1,\ a_k\in\{0,1\}\right\}.$$

For $P(t)=\sum_{k=0}^{n-1}a_ke^{i\omega_kt}$, its Bohr $L^2$ norm satisfies

$$\|P(e^{it})\|_2^2=\int_{\mathbb R}^{\mathrm{Bohr}}\left|\sum_{k=0}^{n-1}a_ke^{i\omega_kt}\right|^2dt=1+\sum_{k=1}^{n-1}a_k^2.$$

Bourgain's conjecture in the Bohr compactification. For any increasing sequence $(\omega_j)_{j\in\mathbb N}$ of real numbers,

$$\sup_{n\geq1}\left\{\sup_{P\in J_{\omega,n}}\frac{\|P(e^{it})\|_1}{\|P(e^{it})\|_2}\right\}<1.$$

This is the formulation of Bourgain's conjecture for frequencies on the Bohr compactification of $\mathbb R$, connecting the polynomial flatness problem with questions about simple Lebesgue spectrum. The source gives no resolution, so the conjecture remains open.