These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of, $${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}\tag1$$ The function $\eta(\tau)$ below is the Dedekind eta function.

I.Case $a=\tfrac13$

*Conjecture:* There are an infinitely many algebraic numbers $\alpha, \beta$ such that
$$H_1(\tau) =\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-\alpha \big)=\beta$$
given by,
$$\alpha = \frac1{4\sqrt{27}}\big(\lambda^3-\sqrt{27}\,\lambda^{-3}\big)^2$$
where $\lambda=\large{\frac{\eta(\frac{\tau+1}3)}{\eta(\tau)}}$ and $\tau=\frac{1+N\sqrt{-3}}2$ for any integer $N>1$. **Examples**:
$$H_1\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-4 \big)=\tfrac3{5^{5/6}}$$
$$H_1\big(\tfrac{1+7\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27 \big)=\tfrac47$$
More details here.

II.Case $a=\tfrac14$

*Conjecture:* There are an infinitely many algebraic numbers $\alpha, \beta$ such that
$$H_2(\tau) =\,_2F_1\big(\tfrac14,\tfrac14;\tfrac34;-\alpha \big)=\beta$$
given by,
$$\alpha = \frac1{4\sqrt{64}}\big(\lambda^6-\sqrt{64}\,\lambda^{-6}\big)^2$$
where $\lambda=\large{\frac{\sqrt2\,\eta(2\tau)}{\zeta_{48}\,\eta(\tau)}},\,$ $\zeta_{48} = e^{2\pi i /48},$ and $\tau=\frac{1+N\sqrt{-1}}2$ for any integer $N>1$. **Examples**:
$$H_2\big(\tfrac{1+3\sqrt{-1}}2)=\,_2F_1\big(\tfrac14,\tfrac14;\tfrac34;-3 \big)=\tfrac2{3^{3/4}}$$
$$H_2\big(\tfrac{1+5\sqrt{-1}}2)=\,_2F_1\big(\tfrac14,\tfrac14;\tfrac34;-80 \big)=\tfrac35$$
More details here.

III.Case $a=\tfrac16$

*Conjecture:* There are an infinitely many algebraic numbers $\alpha, \beta$ such that
$$H_3(\tau) =\,_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-\alpha \big)=\beta$$
with $\alpha$ as a root of the quadratic,
$$16\cdot432\,\alpha(1+\alpha)=-j(\tau)$$
with *j-function* $j(\tau)$ and $\tau = \frac{1+N\sqrt{-3}}{2}$
for any integer $N>1$. **Examples**:
$$H_3\big(\tfrac{1+3\sqrt{-3}}2)=\,_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-\tfrac{125}3 \big)=\tfrac2{3^{5/6}}$$
$$H_3\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9 \big)=\tfrac3{5^{5/6}}\phi^{-1}$$
More details here.

Q:Are the three related conjectures true? And how to prove them? (Bounty is about to expire for the first one.)