Let $\varphi(x_1,\dots, x_k)$ be some smooth function with partial derivatives of magnitude $\asymp 1$ for $x_i\asymp 1$. For concreteness, as it doesn't appear to add much extra structure beyond the general case, one may suppose $\varphi(x_1,\dots,x_k) = \sqrt{x_1\dots x_k}$.
Are there estimates for manyfold iterated exponential sums of the form, say, $$\mathcal S = \sum_{n_1,\dots,n_k\sim X} e\bigg(T\varphi\bigg(\frac{n_1}X,\dots,\frac{n_k}X\bigg)\bigg),$$ which do not diminish if $T$ is a slow growing power of $X$ as $k$ grows? To be precise, is there a slow growing $f(k)$ such that if $X\ll T\ll X^{f(k)}$, we may obtain a bound of the form $\mathcal S\ll X^{k - \delta}$ for some $\delta$ independent of $k$.
