Universal termination conjecture for the proposed algorithm

Let $a$, $b$, and $c$ be pairwise coprime positive integers with $a,c\geq2$. Let $S(a,b,c)$ be a heuristic search bound function and $T(a,b,c,p,k,n)$ an exploration termination function.

Universal termination conjecture. There exist universal functions $S(a,b,c)$ and $T(a,b,c,p,k,n)$ such that, when the proposed algorithm is guided by this pair of functions, it terminates in finitely many steps for every such triple $(a,b,c)$, returns the complete set of positive-integer solutions $(x,y)$ to

$$a^x+b=c^y,$$

and provides a rigorous proof of completeness.

The conjecture formalizes the paper's expectation that its heuristic search and modular-exclusion strategy can be made universally effective. The source presents this as an expectation based on the algorithm's design and empirical success, without providing a proof or resolution.