There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture.
My question is whether there is a set of such conjectures which cannot all be true, eg. is a statement such as "Either $\mathcal{P} = \mathcal{NP}$ or the Riemann hypothesis is wrong" true?
In an early instance of a computer-assisted proof, Douglas Hensley and Ian Richards (Primes in Intervals, Acta Arithmetica 25 (1974), 375-391) proved that two conjectures of Hardy and Littlewood about prime numbers, the first Hardy-Littlewood conjecture (the prime $k$-tuples conjecture, a generalization of the twin primes conjecture) and the second Hardy-Littlewood conjecture (that $\pi(x+y)\le\pi(x)+\pi(y)$), can not both be true. As a result of this work of Hensley and Richards, the second Hardy-Littlewood conjecture has fallen out of favor.