The Alladi-like supernormal identity conjecture

Let $f(n)$ be an arithmetic function, let $\operatorname{idx}(p_n)=n$, let $p_{\operatorname{min}}(n)$ be the prime associated with the least prime factor of $n$, and define $\mu^*(n)=-\mu(n)$. Assume that

$$\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}f(k)=L_f$$

exists. Alladi-like supernormal identity conjecture. Then

$$\sum_{n\geq 2}\frac{\mu^*(n)\left(f\circ\operatorname{idx}\right)\left(p_{\operatorname{min}}(n)\right)}{n}=L_f.$$

The identity is proposed by transferring a partition-theoretic Abelian formula to natural numbers through the paper’s partition–prime analogies. No proof or resolution of this conjectural identity is given.