Moss–Ward conjecture on Dold congruences for quadratic recurrence subsequences

Let $P,Qin\thinspace\mathbb Z$ and let $(u_n)$ be defined by

$$u_{n+2}=Pu_{n+1}+Qu_n\qquad(n\geq 1),$$

with initial conditions $u_0=0$ and $u_1=1$. A sequence satisfies condition (D) when its Dold congruences hold. Moss–Ward conjecture. The sequence

$$\bigl((P^2-4Q)u_{n^2}\bigr)_{n\geq 1}$$

satisfies condition (D). This conjecture concerns the realizability of subsequences of linearly recurrent sequences; the source presents it as an open conjecture proposed by Moss and Ward.