Strong ideological generation of the real numbers by $cmathfrak{W}_{cinfty}$

Let $\text{\usefont{U}{euf}{m}{n}W}_{\infty}$ denote the class of real numbers with the strong ideological generation property. Strong ideological generation conjecture. $\text{\usefont{U}{euf}{m}{n}W}_{\infty}$ strongly ideologically generates $\mathbb R$. This claim concerns the proposed arithmetic structure of Diophantine approximation groups; the supplied text does not indicate whether it has been proved or remains open.