The magic sign-smoothing conjecture for Dirichlet inverse functions

Let $f$ be a non-constant Dirichlet-invertible arithmetic function, with Dirichlet inverse $f^{-1}$. Define the partition sequences

$$p_1(n)=[q^n]\prod_{m\geq 1}(1+q^m),\qquad p_2(n)=[q^n]\prod_{m\geq 1}(1+q^m)^{-1},$$

and the transformations

$$s_1[f](n):=\sum_{j=1}^n f(j)p_1(n-j),\qquad s_2[f](n):=(-1)^n\sum_{j=1}^n f(j)p_2(n-j).$$

Say that $f$ has property $\mathcal{P}_1$ at $N$ if $s_1[f](n)$ has constant sign for all $n\geq N$, and property $\mathcal{P}_2$ at $N$ if $s_2[f](n)$ has constant sign for all $n\geq N$. Define

$$M_1[f]:=\sup\{n\geq 1:f\text{ does not have property }\mathcal{P}_1\text{ at }n\},$$ $$M_2[f]:=\sup\{n\geq 1:f\text{ does not have property }\mathcal{P}_2\text{ at }n\}.$$

Magic sign-smoothing conjecture. If $f(n)$ has constant sign for all $n\geq 1$, then $M_1[f^{-1}]$ and $M_2[f^{-1}]$ are bounded, with eventual signs respectively satisfying

$$\operatorname{sgn}(f(M_1[f^{-1}]))=-\operatorname{sgn}(f(1)),\qquad \operatorname{sgn}(f(M_2[f^{-1}]))=\operatorname{sgn}(f(1)).$$

If $f$ takes both signs, then $M_1[f^{-1}]$ and $M_2[f^{-1}]$ are bounded, with eventual signs respectively satisfying

$$\operatorname{sgn}(f(M_1[f^{-1}]))=\operatorname{sgn}(f(1)),\qquad \operatorname{sgn}(f(M_2[f^{-1}]))=-\operatorname{sgn}(f(1)).$$

The claim predicts eventual sign stabilization after two partition-convolution transformations of Dirichlet inverses; the source gives no proof or resolution.