Transcendence conjecture for strong Engel series with signs

Let $\xi$ be a real number defined by a series of the form

$$$$

for arbitrary $p/q\in\mathbb{Q}$ and positive integer parameters $z_2,z_3,\ldots$. Transcendence conjecture. Every such real number $\xi$ is transcendental. The paper notes that although numbers with irrationality exponent greater than $2$ form a measure-zero set, there is no simple criterion for transcendence when the exponent equals $2$; it therefore expects that none of the real numbers defined by strong Engel series with signs are algebraic.