Are there any well known conjectures that are proved to be provable or falsifiable and yet not proved or falsified? Let me give an example. Consider the statement:
"There is a natural number $x$ which has property $p$".
Now suppose we could prove that, should the number $x$ exist, it must be lower than $10^{10^{10^{10}}}$.
Then it means that we know that the statement can be proved or falsified by checking each number $x$ from $1$ to $10^{10^{10^{10}}}$.
Therefor we have an algorithm for the proof (or an algorithm to falsify the statement). It's just that it would take too long to run. And we don't know the result.
What got me thinking was if we had a statement like "If there is an exceptional zero of the Riemann function then it's imaginary value would have upper limit of $10^{10^{10^{10}}}$", then this might be a proof (or a falsification test) even though it would take too long to run.