Let $q=e^{2\pi it}$. If $u(t)$ is Ramanujan's octic continued fraction, is it true that the generator of the octahedral group can be expressed as a continued fraction of the form
$$
(u(2t))^2=\frac{2q^{1/2}}{1-q+\frac{q(1+q)^2}{1-q^3+\frac{q^2(1+q^2)^2}{1-q^5+\frac{q^3(1+q^3)^2}{1-q^7+\ldots}}}}
$$
for $|q|\lt 1$?