Determine a lower bound for the factorisation algorithm

Determine a lower bound for the computational complexity of deciding whether a matrix in the class [?] can be factored or finding such a factorisation, relative to the algorithmic problem studied for low-dimensional nonnegative integer matrices.

Background

The paper develops an algorithm for matrices in the class $\Xi$, consisting of non-dominated, adj-coprime $2\times2$ nonnegative integer matrices, that determines whether a prescribed determinant $c$ can occur as the determinant of a factor in a nontrivial factorisation $A=BC$. The algorithm runs in soft-linear time $\tilde O(\mu(A))$, where $\mu(A)$ is the smallest entry of $A$.

The authors explicitly state that no lower bound is currently known for this problem. Establishing such a bound would clarify the intrinsic computational difficulty of the factorisation decision and search tasks and would help assess how close the presented algorithm is to optimal.