Is there a variation of Coppersmith’s method that applies to disjoint variable set with additional control?

We have a polynomial $ f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are $ x_1^2,x_1,x_1x_2,x_2,x_2^2,x_3,x_4,x_3x_4$ and we seek solutions $ (x_1,x_2,x_3,x_4)\in\mathbb Z^4$ with $ |x_1|<X_1$ , $ |x_2|<X_2$ , $ |x_3|<X_3$ and $ |x_4|<X_4$ . We see that $ x_1,x_2$ and $ x_3,x_4$ variables do not mix.

It seems the set of $ x_1,x_2$ variables satisfy the generalized lower triangle bound on page $ 16$ http://www.cits.rub.de/imperia/md/content/may/paper/jochemszmay.pdf and the overall set of variables $ x_1,x_2,x_3,x_4$ also satisfy the generalized lower triangle bound.

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In here $ \lambda_1=\lambda_2=\lambda_3=\lambda_4=2$ and $ D=1$ .

Assume we have the additional condition that for a given $ (x_1,x_2)\in\mathbb Z^2$ with $ |x_1|<X_1$ and $ |x_2|<X_2$ we have that there is an unique $ (x_3,x_4)\in\mathbb Z^2$ with $ f(x_1,x_2,x_3,x_4)=0$ , $ |x_3|<X_3$ and $ |x_4|<X_4$ vice versa for a given $ (x_3,x_4)\in\mathbb Z^2$ with $ |x_3|<X_3$ and $ |x_4|<X_4$ we have that there is an unique $ (x_1,x_2)\in\mathbb Z^2$ with $ f(x_1,x_2,x_3,x_4)=0$ , $ |x_1|<X_1$ and $ |x_2|<X_2$ . $ W$ is highest absolute value of coefficient of $ f(x_1X_1,x_2X_2,x_3X_3,x_4X_4)$ .

  1. In this situation can the bound of $ X_1^{\lambda_1}X_2^{\lambda_2}X_3^{\lambda_3}X_4^{\lambda_4}\leq W^\frac1D$ be improved to $ $ \max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3}X_4^{\lambda_4})\leq W^\frac1D$ $ or may be at least $ $ \max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3/2}X_4^{\lambda_4/2})\leq W^\frac2{3D}$ $ ($ \lambda_3/2$ and $ \lambda_4/2$ is based on guess that variables are disjoint and have separate control and $ x_3,x_4$ do not satisfy generalized triangle bound with $ \lambda_3=\lambda_4=1$ and assuming $ x_1,x_2$ variables were not present in given polynomial will give $ W^{\frac2{3D}}$ bound)?

  2. If not what is the best we can do at least for the case $ X_1=X_2=X_3=X_4$ ?

Cross-posted: https://crypto.stackexchange.com/questions/64296/is-there-a-variation-of-coppersmiths-method-that-applies-to-disjoint-variable-s