Critical-limit conjecture for random multiplicative functions

Let $\alpha(n)$ be the random completely multiplicative function appearing in the paper, let $\mu$ be the Möbius function, and let $m_{x,\infty}$ be the random measure defined in the paper. For $f\in\{\mathbf{1},\mu\}$, define

$$V_\infty^{\mathrm{critical}}:=\lim_{x\to\infty}\frac{1}{2\pi}\int_{\mathbb{R}}\frac{\sqrt{\log\log x}\,m_{x,\infty}(ds)}{|\frac12+is|^2}=\frac{1}{2\pi}\int_{\mathbb{R}}\frac{m_\infty^{\mathrm{critical}}(ds)}{|\frac12+is|^2},$$

where $m_\infty^{\mathrm{critical}}(ds):=\lim_{x\to\infty}\sqrt{\log\log x}\,m_{x,\infty}(ds)$ and all limits are in probability. Critical-limit conjecture. The random variable $V_\infty^{\mathrm{critical}}$ is almost surely finite and strictly positive, and, independently of $G\sim\mathcal{N}_{\mathbb{C}}(0,1)$,

$$\frac{(\log\log x)^{1/4}}{\sqrt{x}}\sum_{n\leq x}\alpha(n)f(n)\xrightarrow[x\to\infty]{d}\sqrt{V_\infty^{\mathrm{critical}}}\,G.$$

The convergence is stable, and for every fixed $q\in[0,1)$,

$$\lim_{x\to\infty}\mathbb{E}\left[\left|\frac{(\log\log x)^{1/4}}{\sqrt{x}}\sum_{n\leq x}\alpha(n)f(n)\right|^{2q}\right]=\Gamma(1+q)\mathbb{E}\left[\left(V_\infty^{\mathrm{critical}}\right)^q\right].$$

This refines Harper's question about a nontrivial critical distributional limit. The supplied text presents it as a precise conjecture and gives no resolution, so the existence and properties of the critical limiting random measure remain open.