Non-factorial divisibility for the hypergeometric series h1

Determine whether the denominators of the Taylor coefficients of the hypergeometric power series h1(x)=\sum_{n\ge0}x^n/\prod_{k=0}^{n-1}(k^2+1) can divide an expression of the form \delta^{n+1}(\nu n+\mu)!^s for fixed nonnegative integers \delta,\nu,\mu,s.

Background

The paper considers the hypergeometric series h1(x)=\sum_{n\ge0}x^n/\prod_{k=0}^{n-1}(k^2+1), whose coefficient denominators are d_n=\prod_{k=0}^{n-1}(k^2+1). Unlike the comparison series h2, the associated polynomial has nonrational roots, so the divisibility phenomenon established for split indicial polynomials does not apply.

The authors state that, conjecturally, no fixed factorial-type expression \delta^{n+1}(\nu n+\mu)!^s can account for all d_n. Their explanation invokes the widely believed Bouniakovsky conjecture for values k^2+1, which would produce arbitrarily large prime divisors incompatible with such a fixed factorial bound.