Nonexistence conjecture for rational periodic vectors of type $(L,R,L,R)$

Let $C,D$ be distinct, nonzero rational numbers, and let $f(x,y)=Cx^2+Dy^2$ be a binary quadratic form. A rational periodic vector of type $(L,R,L,R)$ is a rational vector undergoing the corresponding four-step replacement cycle, with left and right entries replaced according to the indicated type. Let $R_{C,D}(y)$ denote the polynomial defined in the paper whose rational roots correspond to such vectors. The $(L,R,L,R)$ nonexistence conjecture. There exist no distinct, nonzero rational numbers $C,D$ such that $f(x,y)=Cx^2+Dy^2$ has a rational periodic vector of type $(L,R,L,R)$; equivalently, $R_{C,D}(y)$ has no rational root. The authors report computational evidence up to height $10^4$, but no proof, and explain that the resulting rational-point problem remains unresolved.