Transcendence degrees of fields generated by exponentials of products

Let $\theta =({\theta }_1,\dots ,{\theta }_m)\in {\mathbb{R}}^m,\kappa =({\kappa }_1,\dots ,{\kappa }_n)\in {\mathbb{R}}^n$ be two tuples of real numbers each linearly independent over $\text{Q}$, and $T$ the transcendence degree of the field generated by $\{\exp ({\theta }_i{\kappa }_j)\mathrm{\mid }i=1,\dots ,m,j=1,\dots ,n\}$ over $\text{Q}$. The estimate $T\ge \frac{mn}{m+n}-1$ has been conjectured for some time but could only be proved under additional hypotheses for $\theta$ and $\kappa$. This paper proves a weaker estimate for $T$ while also reducing the strong estimate to a prominent conjecture on intersections of subvarieties of split tori with subgroups.