Let $c$ be a positive integer and fix $a=c-1$, and $b=c+1$.
Challenge: Find the largest value of $c$ such that $ac\pm1$ and $bc\pm1$ are pairs of twin primes.
For example, with $c=6$ we have $a=5$ and $b=7$ yielding twin primes $5\cdot6\pm1$ (29 and 31) and $6\cdot7\pm1$ (41 and 43).
My conjecture is that if an upper bound for $c$ can be proven, then the twin prime conjecture is false, and if it can be proven that $c$ can be arbitrarily large then an infinite number of twin prime pairs can be generated and thus the twin prime conjecture is true.
Of course if $c$ can be taken arbitrarily large than the twin prime conjecture is true. If there are only finitely many such $c$ then the twin prime conjecture might be true or false. But the smart money says that there are infinitely many such $(a,b,c).$ To support this claim, some examples (giving only $ac-1$):
Standard number-theoretic conjectures suggest that there are infinitely many.