Additive-energy conjecture for lattice points in a three-dimensional thin annulus

For $\lambda>1$ and $\delta\in(0,1)$, let

$$\mathcal{A}_{\lambda,\delta}=\{n\in\mathbb{Z}^3:n_1>|n_2|+|n_3|,\ \lambda-\delta<|n|<\lambda+\delta\}.$$

For an even integer $p$, define the additive energy by

$$E_{p/2}(A)=\left|\left\{(a_1,\ldots,a_p)\in A^p:a_1+\cdots+a_{p/2}=a_{p/2+1}+\cdots+a_p\right\}\right|.$$

Additive-energy conjecture. For $\mathcal{A}_{\lambda,\delta}$,

$$E_{p/2}(\mathcal{A}_{\lambda,\delta})\lesssim\lambda^p\delta^{p/2}+\lambda^{2p-3}\delta^p.$$

This is an additive-combinatorial reformulation of even-integer spectral-projector bounds, with the three-dimensional version presented as an analogue of a conjecture previously stated in dimension two. The supplied text does not state whether this estimate has been proved or disproved.