<s>[INST] Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$ I've found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits.
$${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right) \stackrel{?}{=} \frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$$
Or using a Pfaff transformation in an equivalent form
$$81^{1/6} \cdot {_2F_1}\left(\begin{array}c\tfrac16,\tfrac16\\\tfrac56\end{array}\middle|\,-80\right) \stackrel{?}{=} \frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$$
How could we prove it?

Other related problem: How could we prove that
$${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right) \stackrel{?}{=} {_2F_1}\left(\begin{array}c\tfrac12,\tfrac56\\\tfrac12\end{array}\middle|\,\frac{4}{9}\right) = {_1F_0}\left(\begin{array}c\tfrac56\\\ - \,\end{array}\middle|\,\frac{4}{9}\right)$$
 [/INST] (This may not be complete, but generalizes the observation for context.) The OP's answer mentioned Reshetnikov's integral which can be expressed as,
$$\frac{1}{48^{1/4}\,K(k_3)}\,\int_{-1}^1\frac{dx}{\sqrt[3]{9+4\sqrt{5}\,x}\,\left(1-x^2\right)^{\small2/3} } =\,_2F_1\big(\tfrac{1}{3},\tfrac{1}{3};\tfrac{5}{6};-4\big)= \frac{3}{5^{5/6}} $$
so it will be fruitful to find a transformation between Reshetnikov's use of $\,_2F_1\big(\tfrac{1}{3},\tfrac{1}{3};\tfrac{5}{6};z_1\big)$ and the OP's use of $\,_2F_1\big(\tfrac{1}{6},\tfrac{2}{3};\tfrac{5}{6};z_2\big)$. Employing well-known transformations, we get,

$$G(y)=\,_2F_1\big(\tfrac{1}{3}, \tfrac{1}{3};\tfrac{5}{6}; -y\big) = \big(\tfrac1{1+2y}\big)^{1/3}\,_2F_1\Big(\tfrac{1}{6}, \tfrac{2}{3};\tfrac{5}{6}; \tfrac{(1+2y)^2-1}{(1+2y)^2}\Big)$$

It is conjectured that if $y$ are certain algebraic numbers, than $G(y)$ is also an algebraic number. Specializing the RHS, define as in this post,
$$H(\tau)=\,_2F_1\big(\tfrac16,\tfrac23;\tfrac56;(1-2\delta_4)^2\big),\quad\text{where}\;\frac1{\delta_4}-1=\frac1{27}\left(\tfrac{\eta\big((\tau+1)/3\big)}{\eta(\tau)}\right)^{12}$$
with Dedekind eta function $\eta(\tau)$. If $\tau = \frac{1+N\sqrt{-3}}2$ for some integer $N>1$, then the conjecture implies both $\delta_4$ and $H(\delta_4)$ are algebraic numbers. For example, if we use $\tau = \frac{1+5\sqrt{-3}}2$ and $\tau = \frac{1+7\sqrt{-3}}2$, respectively, then we recover your,
$$H\big(\tfrac{1+\color{blue}5\sqrt{-3}}2\big)=\,_2F_1\big(\tfrac16,\tfrac23;\tfrac56;\tfrac{80}{81}\big) = \tfrac3{\color{blue}5}\,(9\sqrt5)^{1/3}$$
as well as,
$$H\big(\tfrac{1+\color{blue}7\sqrt{-3}}2\big)=\,_2F_1\big(\tfrac16,\tfrac23;\tfrac56;\tfrac{3024}{3025}\big) = \tfrac4{\color{blue}7}\,55^{1/3}$$
and so on.
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