Bourgain's eigenfunction bounds on the three-dimensional torus

Let $u$ be an eigenfunction of the Laplacian $-\Delta$ on $\mathbb{T}^3$ with eigenvalue $\lambda^2$. Bourgain's eigenfunction conjecture. One should have

$$\|u\|_{L^p}\lesssim_\epsilon\begin{cases}\lambda^{\frac12-\frac3p+\epsilon}\|u\|_{L^2}&\text{if }p\geq6,\\lambda^\epsilon\|u\|_{L^2}&\text{if }p\leq6.\end{cases}$$

The loss $\lambda^\epsilon$ might be unnecessary when $p<\infty$. This is the eigenfunction specialization of the spectral-projector problem and concerns sharp $L^p$ growth for toral eigenfunctions. The supplied text does not give a resolution status, so the conjecture remains open here.