conjectured general continued fraction for the quotient of gamma functions

Continued fraction (2) can be simplified as $$ \tan\left(\alpha\tan^{-1}z\right)=\cfrac{\alpha z}{1+\cfrac{\frac{(1^2-\alpha^2)z^2}{1\cdot 3}} {1+\cfrac{\frac{(2^2-\alpha^2)z^2}{3\cdot 5}}{1+\cfrac{\frac{(3^2-\alpha^2)z^2}{5\cdot 7}}{1+\ddots}}}}\tag{2a} $$ This is a special case of the following continued fraction due to N$\ddot{\text{o}}$rlund (B.Berndt, Ramanujan's notebooks, vol.2)

To obtain (2a) from (21.6) set $\beta=-\alpha,\gamma=\frac{1}{2},\frac{x(1-x)}{(1/2-x)^2}=z^2$. Now one can apply the following formulas $$ \, _2F_1\left(a,-a;\frac{1}{2};x\right)=\cos \left(2 a \sin ^{-1}\left(\sqrt{x}\right)\right) $$ $$ -2a^2\, _2F_1\left(1+a,1-a;\frac{3}{2};x\right)=\frac{d}{dx}\left[\, _2F_1\left(a,-a;\frac{1}{2};x\right)\right] $$ $$ 2 \sin ^{-1}\left(\sqrt{x}\right)=\tan^{-1}\frac{\sqrt{x(1-x)}}{1/2-x} $$ to complete the proof.