Generic-parameter extension of the hypergeometrization change-of-variable formula

Let $\mathcal{H}_c^a(x)$ denote the one-parameter group of functions appearing in the change-of-variable formula, and let

$$y=F_m(x):=1-(1-x)^m,$$

with $m\in\mathbb{Z}$. The formula

$$\mathcal{H}_{c}^{a}(x)= \left( \frac{mx}{y} \right)^{1-c}(1-y)^{1+\frac{c-a}{m}}\left( \prod_{j=1}^{m}(1-y)^{\frac{a-c-1}{m}}\mathcal{H}_{c+(j-1)\frac{a-c}{m}}^{c+j\frac{a-c}{m}}(y) \right)\left( \frac{mx}{y} \right)^{a-1}$$

Generic-parameter extension conjecture. The formula holds for generic values of $a,c\in\mathbb{C}$ and all nonzero $m\in\mathbb{Z}$.

The theorem in the paper proves the formula when either $m\in\{-2,-1,1,2\}$ with arbitrary $a,c\in\mathbb{C}$, or when $m\in\mathbb{Z}\setminus\{0\}$ and $a-c\in\mathbb{Z}$. The conjectural extension removes the integer-difference restriction on $a$ and $c$; the source explicitly says that the remaining restrictions reflect only the author's inability to prove the full statement.