Two (probably) equal real numbers which are not proved to be equal?
Thomson’s problem for $n = 7$ provides a nice example:
$$\min_{x_1, \ldots, x_7 \in \mathbb{S}^2} \sum_{i=1}^7 \sum_{j = i + 1}^7 \frac{1}{|x_i - x_j|} = \frac{1}{2} + 10 \frac{1}{\sqrt{2}} + 5\sqrt{\frac{2}{5 + \sqrt{5}}} + 5\sqrt{\frac{1 + \sqrt{5}}{2\sqrt{5}}}$$
Now let me explain. Thomson's problem is to find the minimal energy configuration for $n$ electrons on a unit sphere, i.e. the left-hand-side of the equation. The answer is only rigorously known for $n \leq 6$ and $n = 12$. For $n = 7$ the solution is only conjectured.
Now, the left-hand side of the equation above is computable since it is the minimum of a computable function over the sphere. (Moreover, it is fairly efficient to compute in practice, hence the filled-in table of minimal energies in the Wikipedia article.)
The right hand side is the energy of the conjectured optimal configuration for the $n=7$ case. (See here for the specific distances between points, which I used in the above equation.)
Many of the other values of $n$ have conjectured solutions and therefore give rise to similar equations.
One thing which I find crazy about this example (and which might ruin it for you), is that while this equation has never been rigorously proved, it’s truth/falsity is decidable by the decidability of real-closed fields (see my answer to a similar problem here). Hence a clever programmer-mathematician may find a computer-assisted rigorous proof. Indeed, this is exactly what happened for the $n=5$ case. (Schwartz used custom code. It would be nice to have someone redo this computation in a formal theorem prover like HOL Light, Coq, or Lean.)
As mentioned in another MO question, Gourevitch's conjecture is a nice example: $$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$
The conjectures on special values of $L$-functions provide a lot of examples. For example, David Boyd conjectured in his celebrated paper that the Mahler measure of the Laurent polynomial $P_k(x,y)=x+\frac{1}{x}+y+\frac{1}{y}+k$, where $k$ is an integer $\neq 0, \pm 4$, is proportional to $L'(E_k,0)$, where $E_k$ is the elliptic curve defined by the equation $P_k(x,y)=0$. It is not difficult to check these identities numerically to thousands of decimal places, but so far they have been proved only in a finite (and small) number of cases. (Technically you asked about equalities, here they involve some rational factor, which is however simple enough to guess in each particular case, although its value in general is mysterious, e.g. may be linked to the Bloch-Kato conjectures).
The equality mentioned in the OP (page 10 of Bailey-Borwein-Broadhurst-Zudilin) is essentially Borel's theorem in disguise for the $K$-group $K_3(\mathbb{Q}(\sqrt{-7}))$. Equation (10) of the same article is an instance of the following question: we have two elements in some $K$-group which has (or should have) rank 1, so they should be proportional hence their regulators should also be proportional. In the present case, equation (10) should follow from the 5-term functional equation of the dilogarithm evaluated at particular algebraic arguments, which is however not an easy task (there is an obvious but inefficient algorithm since the set of algebraic numbers is countable).