Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal?


I never really understood the assertion that "the reals do not have decidable equality" until I saw concrete examples such as [this gem](https://mathoverflow.net/a/36250/1384). It's clear that both sides are well-defined real numbers, however it's not at all clear how one might go about proving their equality. However, perhaps a little disappointingly, on page 10 of [this article](https://arxiv.org/abs/1005.0414) by Bailey, Borwein, Broadhust and Zudilin, the equality is proved (and that article appeared on ArXiv before Stanley's post). 


It's very easy to give "silly" examples of real numbers which are probably equal but not provably equal, e.g. the real number which is 0 if the Birch and Swinnerton-Dyer conjecture is true and 1 if not, is probably equal to zero, but we can't prove it. So for the pedants, can anyone give me an example of two computable real numbers whose equality is believed, but not known? More informally, I am looking for an example like the one Richard Stanley mentions in the link above where computer scientists can compute both sides to a gazillion decimal places but mathematicians can't prove equality rigorously? Stanley calls such things rare. Several potential examples are mentioned on p13 of the BBBZ paper I cited above, however that paper was written ten years ago and things seem to move fast in this area. Are any of these still open? It looks to me like the authors are developing techniques which might solve them. 
For example equation (10): if


$$Cl\_2(\theta):=-\int\_0^\theta\log|2\sin(\sigma)|d\sigma$$


and $\theta\_2:=2\tan^{-1}(\sqrt{2})$ then is


$$27Cl\_2(\theta\_2)−9Cl\_2(2\theta\_2) + Cl\_2(3\theta\_2)= 8Cl\_2(\pi/4)+ 8Cl\_2(3\pi/4)$$
still an open problem?



Two (probably) equal real numbers which are not proved to be equal?
As mentioned in [another MO question](https://mathoverflow.net/a/100290/3106),
[Gourevitch's conjecture](https://doi.org/10.1080/10586458.2003.10504518) is a nice example:
$$\sum\_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$


