The indicator-function residue conjecture for prime tower factorizations

Let $q$ be the parameter defining the set $M^c(q)$, let $d(q)$ denote the density associated with this set, and let $I(n)$ be the indicator function of the event $n\in M^c(q)$. Define

$$f(s)=\sum_{m=1}^{\infty}I(m)m^{-s},$$

where $f(s)$ is understood as the analytic continuation of this Dirichlet series. Residue conjecture. The density complement satisfies

$$1-d(q)=\prod_{p\ \operatorname{prime}}\left[\lim_{s\to1^+}\left(1-\dfrac{1}{p^s}\right)\sum_{m=0}^{\infty}\dfrac{I(p^m)}{p^{ms}}\right]=\lim_{s\to1^+}\prod_{p\ \operatorname{prime}}\left[\left(1-\dfrac{1}{p^s}\right)\sum_{m=0}^{\infty}\dfrac{I(p^m)}{p^{ms}}\right]$$

and

$$1-d(q)=\lim_{s\to1^+}\dfrac{1}{\zeta(s)}\sum_{m=1}^{\infty}\dfrac{I(m)}{m^s}=\lim_{s\to1^+}(1-s)\sum_{m=1}^{\infty}\dfrac{I(m)}{m^s}=\operatorname{Res}_{s=1}f(s).$$

The formula is intended to connect the density of integers excluded from the prime-tower set with the Euler product and the residue of the associated Dirichlet series; the authors explicitly leave the analytic details unproved, and no resolution is supplied.