Using a numerical search on my computer I discovered the following inequality:
$$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$
where $\rho$ is the positive root of the polynomial equation
$$12\,\rho^8-12\,\rho^4-8\,\rho^2-1=0,\tag2$$
that can be expressed in radicals:
$$\rho=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]{4\vphantom{\large1}}}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}.\tag3$$
Based on this inequality I conjecture that the actual difference is the exact zero, i.e.
$$\color{#808080}{_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\rho}.\tag4$$
I looked up in DLMF and MathWorld, but did not find a known special value with exactly these parameters. It also appears that CAS like *Maple* or *Mathematica* do not know this identity.

Could you please suggest any ideas how to prove the conjecture $(4)$?

*Update:* I can propose even more general conjecture:
$$\color{#808080}{27\,(x-1)^2\cdot{_2F_1}\left(\tfrac14,\tfrac34;\tfrac23;x\right)^8+18\,(x-1)\cdot{_2F_1}\left(\tfrac14,\tfrac34;\tfrac23;x\right)^4-8\cdot{_2F_1}\left(\tfrac14,\tfrac34;\tfrac23;x\right)^2=1}$$