Four exponentials conjecture

The four exponentials conjecture asserts that if $x_1,x_2$ are complex numbers linearly independent over $\mathbb{Q}$, and $y_1,y_2$ are likewise linearly independent over $\mathbb{Q}$, then at least one of the four numbers $e^{x_i{y}_j}$ is a transcendental number.

Both independence hypotheses are essential to the statement. It is a problem in transcendental number theory, asking how a two-by-two array of products constrains exponentiation. The analogous statement with two independent $x_i$ and three independent $y_j$ is the six exponentials theorem; that stronger supply of inputs is known to force at least one transcendental output. The four-number assertion is a conjecture, not a consequence of checking individual exponentials.