The S-number conjecture for E- and M-function values

Let ${\bf E}$ and ${\bf M}$ denote the sets of values under consideration of $E$-functions and $M$-functions, respectively. A complex number is an $S$-number when its Mahler exponent satisfies $0<w(\xi)<\infty$. The S-number conjecture. Every transcendental element of ${\bf E}\cup {\bf M}$ is an $S$-number. This is presented as a general heuristic for transcendental elements of ${\bf E}$ and ${\bf M}$; the context notes that the conjecture is known for values of $M$-functions satisfying possibly inhomogeneous first-order equations, while the general statement remains open.