The short interval $L^4$ conjecture for square frequencies

For $N$ large and $\alpha<1$, consider trigonometric polynomials whose frequencies lie in

$$\{N^2,\ldots,(N+N^\alpha)^2\}.$$

The short interval conjecture. For every $\alpha<1$, every such trigonometric polynomial satisfies

$$\|f\|_4\ll_\alpha\|f\|_2.$$

The source says this is trivial for $\alpha\leq1/2$ and completely open for $\alpha>1/2$, with a related stronger conjecture involving logarithmic intervals.