Finite-pole conjecture for the Frobenius matrix

The Frobenius matrix $F(\Gamma)$ is the matrix arising from the Frobenius action in the recursive computation of the zeta function, with parameter $\Gamma$. At infinity, a function has a pole of finite order if it is meromorphic there with a pole of bounded finite order, rather than having an essential singularity. Finite-pole conjecture. The Frobenius matrix $F(\Gamma)$ has a pole of finite order at infinity, rather than an essential singularity. This conjecture would ensure that the Frobenius matrix has controlled behavior at infinity, supporting the effectiveness of the recursive method for computing zeta functions. The supplied text does not state whether the conjecture has been proved or disproved.