Conjectural prime-local denominator bounds at Gompertz factorial endpoints

Assume that the real Borel-summed Euler–Gompertz constant $\delta$ is algebraic. Conjecturally there is $C\ge1$ such that for every $n$ there is a positive integer $D\le C^{n+1}$ for which $Dq_n(\delta)$ is integral at every finite prime: for each prime $p$, some integer $d$ coprime to $p$ makes $dDq_n(\delta)$ an algebraic integer. The proved local-to-global and recurrence lemmas recover the original simultaneous denominator bound. This is an equivalent local arithmetic formulation, not a proof or a weaker logical conjecture.