Identify the function defined by the half-shifted small Apéry continuous generalisation

Determine or characterise the function f(z), defined up to a Möbius transformation, by the continued fraction obtained from Theorem 6.3 via the half-shift n → n + 1/2, namely the continued fraction with partial denominators 1, 12(1 − z^2), and 44 n^2 + 1 + 36 z^2 (for n ≥ 1) and partial numerators 60 z^2 and ((2n + 1)^2 + 16 z^2)((2n + 1)^2 + 36 z^2) (for n ≥ 1), which at z = 0 reduces to the continued fraction for (Γ(1/4)/Γ(3/4))^4 from Theorem 2.1.

Background

Starting from the continuous generalisation of small Apéry in Theorem 6.3, the authors consider the half-shift n → n + 1/2 applied to that continued fraction and define a function f(z) from the resulting expression.

They observe that f(z) is only defined up to a Möbius transformation and provide several empirical properties (e.g., evenness, special values, asymptotics), but explicitly pose the question of identifying this function. The question emphasizes whether there is a recognisable closed-form or standard special-function description for f(z).