Dold-condition conjecture for discriminant-scaled Lucas sequences along square indices

For integers $P,Q$, define the Lucas sequence $(U_n(P,Q))$ by

$$\frac{x}{1-Px+Qx^2}=\sum_{n=0}^{\infty}U_n(P,Q)x^n.$$

The sequence is said to satisfy the Dold condition when its associated Dold congruences hold. Lucas-sequence conjecture. For every $P,Q\in\mathbb{Z}$, the sequence

$$\bigl((P^2-4Q)U_{n^2}(P,Q)\bigr)$$

satisfies the Dold condition.

The conjecture extends the paper's established realizability result for $(5U_{n^2}(1,-1))$ and is motivated by numerical experiments; no resolution is given here.