The conjectural mean value estimate for Weyl sums associated with $(n^3,n)$

For $0\leq u\leq d-1$, define

$$\mathcal{I}_{p,d}(u;N)=\int_{[0,1)\times[0,N^{-u})\times[0,1)^{d-2}}|f_d(\boldsymbol{\alpha};N)|^p\,d\boldsymbol{\alpha},$$

where $d\geq2$, $\boldsymbol{\alpha}=(\alpha_d,\ldots,\alpha_1)\in\mathbb{R}^d$, and

$$f_d(\boldsymbol{\alpha};N)=\sum_{1\leq n\leq N}e(\alpha_dn^d+\cdots+\alpha_1n).$$

The conjectural Weyl-sum estimate. For $d\geq2$, $0\leq u\leq d-1$, and even $p$, one has

$$\int_{[0,N^{-u})\times[0,1)^{d-1}}|f_d(\boldsymbol{\alpha};N)|^p\,d\boldsymbol{\alpha}\ll N^{\epsilon}\left(N^{p-d(d+1)/2}+N^{p/2-u}\right).$$

In the surrounding discussion, the case $d=3$ and $u=2$ is related to the conjectural mean value estimate for Weyl sums associated with $(n^3,n)$, and is stated to remain open; the case $u=1$ is an intermediate result proved by the paper's corollary. The source does not provide a resolution of the general conjecture.