A classical result of Fouvry and Iwaniec states that if $\alpha_1,\ldots, \alpha_4$ are nonzero, $M_1,\ldots, M_4 \geq 1$, $X > 0$, and $|\varphi_{m_1,m_2}|,|\psi_{m_3,m_4}|\leq 1$ are complex numbers, then \begin{align} \tag{1} \sum_{m_j\sim M_j} \varphi_{m_1,m_2} &\psi_{m_3,m_4} e\left(X \frac{m_1^{\alpha_1}m_2^{\alpha_2}m_3^{\alpha_3}m_4^{\alpha_4}}{M_1^{\alpha_1}M_2^{\alpha_2}M_3^{\alpha_3}M_4^{\alpha_4}} \right) \\ &\ll (M_1M_2M_3M_4)^{1+\varepsilon} \left(\left(\frac{X}{M_1M_2M_3M_4}\right)^{1/2} + \frac{1}{(M_1M_2)^{1/2}} + \frac{1}{(M_3M_4)^{1/2}} + \frac{1}{X^{1/2}}\right) . \end{align} Here the notation $m_j\sim M_j$ means $M_j < m_j \leq 2M_j$.
Robert and Sargos proved a similar bound for 3D sums. With the same assumptions as before, and also $\alpha_3 \neq 1$, \begin{align} \tag{2} \sum_{m_j\sim M_j} \varphi_{m_1,m_2} &\psi_{m_3} e\left(X \frac{m_1^{\alpha_1}m_2^{\alpha_2}m_3^{\alpha_3}}{M_1^{\alpha_1}M_2^{\alpha_2}M_3^{\alpha_3}} \right) \\ &\ll (M_1M_2M_3)^{1+\varepsilon} \left(\left(\frac{X}{M_1M_2M_3^2}\right)^{1/4} + \frac{1}{(M_1M_2)^{1/4}} + \frac{1}{M_3^{1/2}} + \frac{1}{X^{1/2}}\right). \end{align} The exponents in each term of (2) are conjecturally sharp. At the very least, (2) gives the optimal bound for the error term in the problem of counting finite abelian groups (e.g. Theorem 4 of the cited paper of Robert and Sargos). My question is, are the exponents in the terms on the right of (1) known/conjectured to be sharp? That is, for each of the four terms on the right of (1), is there a choice of the coefficients $\varphi,\psi$ and exponents $\alpha_j$ such that the $\ll$ can be replaced by $\gg$ and the remaining 3 terms deleted? For example, are there choices of $\varphi,\psi,\alpha_j$ such that $$ \sum_{m_j\sim M_j} \varphi_{m_1,m_2} \psi_{m_3,m_4} e\left(X \frac{m_1^{\alpha_1}m_2^{\alpha_2}m_3^{\alpha_3}m_4^{\alpha_4}}{M_1^{\alpha_1}M_2^{\alpha_2}M_3^{\alpha_3}M_4^{\alpha_4}} \right) \gg(xM_1M_2M_3M_4)^{1/2}? $$ Any help/references are most appreciated.
References:
Fouvry, Etienne; Iwaniec, Henryk, Exponential sums with monomials, J. Number Theory 33, No. 3, 311-333 (1989). ZBL0687.10028.
Robert, O.; Sargos, P., Three-dimensional exponential sums with monomials, J. Reine Angew. Math. 591, 1-20 (2006). ZBL1165.11067.