Logarithmic \(L^6\) Strichartz estimate for the circle Schrödinger propagator

Let \(e^{it\partial_x^2}\) be the free Schrödinger propagator on the circle \(\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})\). Is it true that there exists a constant \(C>0\) such that, for all sufficiently large \(N\) and all \(f\in C^\infty(\mathbb{T})\) satisfying \(\operatorname{supp}\widehat{f}\subset[-N,N]\), the estimate

\[ \|e^{it\partial_x^2}f\|_{L^6_{t,x}([0,2\pi]\times\mathbb{T})} \le C(\log N)^{1/6}\|f\|_{L^2(\mathbb{T})} \]

holds?

Let \(e^{it\partial_x^2}\) be the free Schr\"odinger propagator on the circle \(\mathbb T=\mathbb R/(2\pi\mathbb Z)\). Is it true that there exists a constant \(C>0\) such that, for all sufficiently large \(N\) and all \(f\in C^\infty(\mathbb T)\) satisfying \(\operatorname{supp}\widehat f\subset[-N,N]\), the estimate
\[
  \|e^{it\partial_x^2}f\|_{L^6_{t,x}([0,2\pi]\times\mathbb T)}
  \le
  C(\log N)^{1/6}\|f\|_{L^2(\mathbb T)}
\]
holds?