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 unefficient algorithm since the set of algebraic numbers is countable).

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).