Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1

Question

Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent?

1. $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$.
2. There exists an increasing sequence ${n_k}$ such that $\lim_{k\rightarrow\infty}\hat{\mu}(n_k)=1$.