Nonharmonic Fourier series without the Riemann–Lebesgue property

We prove that in the class of separated sequences $\lambda_n$ there exists a sequence whose real parts decrease arbitrarily slowly to $-\infty$, so that for some continuous function $f$ on $[0,1]$ the general term of the nonharmonic Fourier series $f(t)\sim\sum c_ne^{\lambda_nt}$ diverges to infinity as $n=n_k\to\infty$ for all $t\in(0,1)$.