Extension conjecture for residue functions and adjoint ideals

Let $X$ be the complex manifold, $L$ a line bundle, $\psi$ the function, $\delta$ the constant in the curvature assumption, and $m_k$ the indexed multiplier-ideal data used above. Write $\operatorname{lc}(m_k)$ for the corresponding log-canonical locus and let $\operatorname{Adj}(m_k)$ and $\operatorname{Adj}^{-1}(m_k)$ denote the adjoint-ideal sheaves appearing in the statement. For a residue function $g$, let $\mathcal{R}(g)(\varepsilon;\ell)[\operatorname{lc}(m_k),\sigma]$ denote the residue norm defined above.

Conjecture 1.1.3. Under the curvature assumption given in Theorem, there exists a sufficiently large constant $\ell\geq e$, depending only on $\psi$ and $\delta$, such that for every integer $\sigma\geq1$ and every

$$f\in H^0\mathopen{}\mathclose\bgroup\originalleft(\operatorname{lc}(m_k),K_X\otimes L\otimes\frac{\operatorname{Adj}(m_k)}{\operatorname{Adj}^{-1}(m_k)}\aftergroup\egroup\originalright),$$

there is a holomorphic extension

$$F\in H^0\mathopen{}\mathclose\bgroup\originalleft(X,K_X\otimes L\otimes\operatorname{Adj}(m_k)\aftergroup\egroup\originalright)$$

satisfying

$$\mathcal{R}(F)(1;\ell)[\operatorname{lc}(m_k),\sigma]\leq\mathcal{R}(f)(0)[\operatorname{lc}(m_k),\sigma].$$

The conjecture seeks a uniform residue-norm extension theorem under the stated curvature hypothesis; the supplied text gives no resolution status, so it remains open.