Can a Single Equation Witness that every r.e. Set Admits a Finite-fold Diophantine Representation?

As of today, the question remains open as to whether the quaternary quartic equation 9 \ub7 (u^2 + 7 v^2)^2 12 7 \ub7 (r^2 + 7 s^2)^2 = 2 , (*) which M. Davis put forward in 1968, has only finitely many solutions in integers. If the answer were affirmative then\u2014as noted by M. Davis, Yu. V. Matiyasevich, and J. Robinson in 1976\u2014every r.e. set would turn out to admit a single-fold polynomial Diophantine representation. New candidate \u2018rule-them-all\u2019 equations, constructed by the same recipe which led to (*), are proposed in this paper