Guillera–Almkvist conjecture on rationality and algebraicity in Ramanujan-like series

Let $q=|q|e^{i\pi r}$, and use $\alpha_0=\alpha(q_0)$, $\alpha_1=\alpha(q_1)$, $\tau_0=\tau(q_0)$, and analogous notation for the other quantities. Suppose

$$\sum_{n=0}^{\infty} z(q)^n \left[ \prod_{i=0}^4 \frac{(s_i)_n}{(1)_n} \right] (a(q)+b(q)n+c(q)n^2)=\frac{1}{\pi^2}.$$

Guillera–Almkvist conjecture. The quantities $r$, $\alpha_0$, and $\tau_0^2$ are rational if and only if $z_0$, $c_0$, $b_0$, and $a_0$ are algebraic.

This conjecture was first stated for hypergeometric series and later extended to all cases considered here. The supplied text gives no resolution status.