Determine the constant C(z) in the asymptotic error for the small Apéry continuous generalisation

Ascertain a closed-form expression for the constant C(z) appearing in the asymptotic speed of convergence in Theorem 6.3 for the continued fraction representation of (cosh(π z) − 1)/(3 z^2), and in particular determine C(1).

Background

Theorem 6.3 provides a continuous generalisation of the small Apéry continued fraction, yielding a continued fraction for (cosh(π z) − 1)/(3 z^2) with a quantified speed of convergence. The asymptotic includes a multiplicative constant C(z) that is independent of n.

The authors explicitly state they have not been able to recognise this constant C(z), even at the specific value z = 1, indicating an unresolved identification problem tied to the asymptotics of the continued fraction approximants.