The mixed-norm Strichartz conjecture on the real line times the circle

Let $N\geq 1$, let $\varphi$ be the time cutoff appearing in the estimate, and let $\phi\in L^2(\mathbb{R}\times\mathbb{T})$. For $\xi_0\in\mathbb{R}\times\mathbb{Z}$, write $|\xi-\xi_0|$ for the Euclidean distance in frequency space. The mixed-norm Strichartz conjecture. For all $p\geq 2$, one has

\sup_{\xi_0\in\mathbb{R}\times\mathbb{Z}}\left\\|\varphi(t)\int_{ \substack{\xi\in\mathbb{R} \times \mathbb{Z} \mid \xi-\xi_0\mid\leq N}} e^{ix_1\cdot \xi_1-it\mid\xi\mid^2} \widehat{\phi}(\xi)\\ {\rm d}\xi\right\\|_{L^p_{t,x_1}(\mathbb{R}\times\mathbb{R})} \lesssim (N^{1-\frac{3}{p}}+1) \\|\phi\\|_{L^2(\mathbb{R}\times\mathbb{T})}.

The estimate would describe the pointwise behavior of the linear Schrödinger flow on $\mathbb{R}\times\mathbb{T}$ despite the lack of dispersion in the compact factor. It is particularly challenging because the corresponding dispersive estimates fail on the product space; the conjecture is presented as an open problem.