Strichartz conjecture for the nonelliptic Schrödinger equation on irrational tori

Let $\tilde{\Delta}:=\frac{1}{2\pi}(\partial_{11}+\alpha\partial_{22})$ on $\mathbb{T}^2$, where $\alpha<0$ is irrational, and let $e^{it\tilde{\Delta}}f$ denote the solution to $i\partial_t v-\tilde{\Delta}v=0$ with initial data $f$. Suppose that the Fourier support of $f$ is contained in $[-N,N]^2$. The Strichartz conjecture. For irrational $\alpha$, one has

$$\|e^{it\tilde{\Delta}}f\|_{L^p([0,T]\times\mathbb{T}^2)}\leq C_{\epsilon,\alpha}N^{\epsilon}\|f\|_{L^2(\mathbb{T}^2)}\begin{cases}T^{\frac1p}&\text{for }2\leq p\leq4,\\ T^{\frac1p}+N^{1-\frac4p}&\text{for }4\leq p\leq6,\\ T^{\frac1p}N^{1-\frac6p}+N^{1-\frac4p}&\text{for }6\leq p. \end{cases}$$

The range $2\leq p\leq4$ is known when $T\geq N$, while the conjecture remains open in the other stated ranges. It is an analogue of the corresponding conjecture for the elliptic Schrödinger equation on irrational tori.