A uniform metrical theorem in multiplicative Diophantine approximation

For Lebesgue generic $(x_1,x_2)\in {\mathbb{R}}^2$, we investigate the distribution of small values of products $q\cdot \mathrm{\parallel }q{x}_1\mathrm{\parallel }\cdot \mathrm{\parallel }q{x}_2\mathrm{\parallel }$ with $q\in \mathbb{N}$, where $\mathrm{\parallel }\cdot \mathrm{\parallel }$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q\le T$ such that $$a_T<q\cdot \mathrm{\parallel }q{x}_1\mathrm{\parallel }\cdot \mathrm{\parallel }q{x}_2\mathrm{\parallel }\le b_T\text{and}\mathrm{\parallel }q{x}_1\mathrm{\parallel },\mathrm{\parallel }q{x}_2\mathrm{\parallel }\le c_T$$ for given sequences $a_T,b_T,c_T$ satisfying certain growth conditions.