Let $a\_n$ denote the Fibonacci numbers, for a prime $p$ let $\alpha(p)$ denote the first index $n$ such that $p|a\_n$ and let $r$ denote the golden ratio.


Q: Is there a proof of $\lim\_{x\rightarrow \infty}\frac{1}{x^2}\sum\_{\alpha(p)\leq x} \log p = \frac{3 \log r}{\pi^2}$ ?


The sum is taken over all primes $p$ that divide at least one Fibonacci number with index less or equal to $x$. 


It is related to the number of Wall-Sun-Sun primes (cf. e.g. [Is the Crandall, Dilcher and Pomerance heuristic concerning Wall-Sun-Sun primes still state of the art?](https://mathoverflow.net/questions/76205/is-the-crandall-dilcher-and-pomerance-heuristic-concerning-wall-sun-sun-primes)).


Added later: It is known (P. Kiss, Primitive Divisors of Lucas Numbers, "Application of Fibonacci Numbers" (A. N. Phillipou et al.,Ed.), pp. 29 -- 38, Kluwer Acad. Publ. 1988) that $\limsup\_{x\rightarrow \infty}\frac{1}{x^2}\sum\_{\alpha(p)\leq x} \log p \leq \frac{3 \log r}{\pi^2}$.



Sum of the log of all primes dividing at least one Fibonacci number up to index x
As far as I understand the answer to my question is no. That seems plausible since, differently from what I thought first, the truth of the conjecture apparently does not have serious consequences. 


