Iterated orthogonal extension conjecture from log-canonical centres

Let $X$ be the space and let $\vphi_L$, $\psi$, $\ell$, $\spH$, $\spE$, $\sigma$, $\sigma_{\operatorname{mlc}}$, $\lcc$, $\multidl_{}\paren{\vphi_L}$, and $\RTF|\cdot|(t)[\sigma]$ be as in the paper. Assume that $\spH=\spH[\sigma]\oplus\spE$ is the orthogonal decomposition with respect to the squared norm $\RTF|\cdot|(1)[\sigma]$. Iterated orthogonal extension conjecture. There exists a constant $b\geq 1$, independent of $\vphi_L$ and $\psi$, such that, after normalising $\log\abs{\ell\psi}\geq b$ on $X$, for every $f\in\spH[\sigma_{\operatorname{mlc}}+1]/\spH[1]$ one can inductively find $F_\sigma\in\spE$, for $\sigma=1,\dots,\sigma_{\operatorname{mlc}}$, beginning with $\sigma=\sigma_{\operatorname{mlc}}$, so that for every such $\sigma$,

$$\sum_{j=\sigma}^{\sigma_{\operatorname{mlc}}}F_j\equiv f\pmod{\multidl_{}\paren{\vphi_L}\cdot\mathcal{I}_{\lcc}}\quad\text{on }X,$$

and

$$\RTF|F_\sigma|(1)[\sigma]\leq\RTF|F_\sigma|(0)[\sigma]=\RTF|f-\sum_{j=\sigma+1}^{\sigma_{\operatorname{mlc}}}F_j|(0)[\sigma].$$

Consequently, $F=\sum_{\sigma=1}^{\sigma_{\operatorname{mlc}}}F_\sigma$ is a holomorphic extension of $f$ and, if additionally $\abs\psi\geq1$ on $X$, satisfies

$$\RTF|F|(1)[\sigma_{\operatorname{mlc}}]\leq\sum_{\sigma=1}^{\sigma_{\operatorname{mlc}}}\RTF|F_\sigma|(1)[\sigma]\leq\sum_{\sigma=1}^{\sigma_{\operatorname{mlc}}}\RTF|F_\sigma|(0)[\sigma].$$

The conjecture proposes an iterated extension procedure from log-canonical centres with an $L^2$ estimate; its resolution is not indicated in the supplied text.