The general Clausen identity for a bivariate hypergeometric function

Let $a$, $b$, and $c$ be general complex parameters, and let ${}_{2}\operatorname{F}_{1}$ and $F^{2:1;1}_{1:1;1}$ denote the hypergeometric functions appearing below. Generalized Clausen identity. The following identity should hold:

$$\frac{\Gamma\left(\frac12\right)\Gamma\left(a+b+\frac12\right)\Gamma(1-c)\,\Gamma(1+a+b-c)}{\Gamma\left(a+\frac12\right)\Gamma\left(b+\frac12\right)\Gamma(1+a-c)\,\Gamma(1+b-c)}\,{}_{2}\operatorname{F}_{1}\left(\left.{a,\,b\atop c}\right|z\right)^2= F^{2:1;1}_{1:1;1}\left(\left.{2a;\,2b;\,c-\frac12,\,a+b-c+\frac12\atop a+b+\frac12;\,2c-1,\,2a+2b-2c+1}\right|z,1-z\right).$$

This is proposed as a two-variable generalization of Clausen's identity, relating the square of a Gauss hypergeometric function to a bivariate hypergeometric series. The source provides heuristic support from symmetric-square differential equations and special terminating cases, but does not establish the identity for general parameters.