Bourgain's sharp logarithmic Strichartz bound on the two-dimensional torus

Let $P_N$ denote the frequency projection onto frequencies $\lesssim N$, and let $e^{it\Delta_{\mathbb{T}^2}}$ be the Schrödinger propagator on $\mathbb{T}^2$. For all sufficiently large $N$ and every $\phi\in L^2(\mathbb{T}^2)$, the sharp logarithmic Strichartz conjecture.

\left\\|e^{it\Delta_{\mathbb{T}^2}}P_N\phi\right\\|_{L^4_{t,x}([0,1]\times\mathbb{T}^2)}\lesssim (\log N)^{1/4}\left\\|\widehat{\phi}\right\\|_{\ell^2(\mathbb{Z}^2)}.

Bourgain's upper bound is $N^\epsilon$ for every positive $\epsilon$, while a lower bound of order $(\log N)^{1/4}$ is known. The conjecture asserts that this lower bound gives the optimal growth, up to a constant factor, and is motivated by the corresponding trilinear estimates in the paper.