Stable martingale central limit theorem for multiplicative functions with prime variance parameter

Let $f\in\mathcal{M}$. Suppose that

$$\sum_{p\leq t}|f(p)|^2=\theta\frac{t}{\log t}+O\left(\frac{t}{\log^2 t}\right)$$

for some $\theta\in(0,1)$, and that there exists $c>0$ such that $|f(p^k)|^2=O(2^{k(1-c)})$ and $|f(p)|<1/c$ for all $k\geq 2$ and primes $p$. The conjecture. Under these conditions, the stable convergence in the distributional limit defined by

$$holds, and the moment convergence in$$

holds for every fixed $q\in[0,1/\theta)$. This extends the expected range of the parameter $\theta$ beyond the technical range available from the paper's $L^2$ approximation; the source does not establish the claim for this full range.