Extension of the lognormal multifractal limit theorem

Let $M$ be the lognormal multifractal random measure with parameter $\gamma^2$, and let the extended singular integral be defined, for $x\in[0,1]$ and continuous $f$, by

$$\int_0^1 \left(\frac{\tau}{|t-x|}\right)_+^{\gamma^2}f(t)M(dt)=\lim_{\eta\to0}\int_{t\in[0,1];\,|t-x|>\eta}\left(\frac{\tau}{|t-x|}\right)_+^{\gamma^2}f(t)M(dt).$$

Let $\mu_z^2$ and $K_z$ denote the unique constant and bounded function characterized by the limiting equations of Theorem 3, and let $\upsilon$ be the probability measure whose Stieltjes transform is $\mu_z^2$. Extension conjecture. Theorem 3 should hold in the lognormal multifractal case for all $\gamma^2\in[0,2[$, and the limiting equations should be obtained from those of the regularized theorem with $2W=\omega_\epsilon$ as $\epsilon\to0$.

This would extend the main limiting characterization beyond the small-intermittency range required by the paper's proofs, using the extended definition of the singular integral.