Near-square classification conjecture for binary recurrence sequences

Let $a$ and $b_{1}$ be relatively prime positive integers and put $b=-b_{1}^{2}$. Define the binary recurrence sequence by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$, and $u_{n+2}(a,b)=a u_{n+1}(a,b)+b u_{n}(a,b)$. Let $\mathrm{S}=\{x^{2}:x\in\mathbb{Z}\}$ and $c\mathrm{S}=\{cx:x\in\mathrm{S}\}$. Near-square classification conjecture. For $n>8$ and $|c|=1$ or $|c|$ prime, one has $u_{n}(a,b)\in c\mathrm{S}$ if and only if $(n,a,b,c)=(11,4,-25,3719)$ or $(13,3,-4,181)$. This conjecture seeks a uniform classification of terms in binary recurrence sequences that are a square or a prime multiple of a square; the stated exceptional quadruples are the only cases asserted by the conjecture.