Transcendence conjecture for the continued fraction F(x,y)

Let $x$ and $y$ be integers, and define the continued fraction

$$F(x,y)=[x,y^{-1},x^{2},y^{-2},x^{3},y^{-3},x^{4},y^{-4},\dots].$$

Transcendence conjecture. If $x$ and $y$ are positive integers with $x\neq y$, then $F(x,y)$ is transcendental.

This conjecture concerns a non-periodic, non-simple continued fraction arising from two interlacing geometric series. The source presents it as suggested by preliminary computer experiments; no resolution is given there.