Lower semi-continuity conjecture for log plurigenera of proper SNCL schemes

Let $\kappa$ be a field, let $s$ be the log point of $\kappa$, and let $X/s$ be a proper SNCL scheme of pure dimension $d$. For $n,r\in\mathbb Z_{\geq 1}$, define the log geometric plurigenus in positive characteristic by

$$p_g(X/s,n,r):={\rm length}_{{\cal W}_n}H^0\left(X,({\cal W}_n\Omega^d_X)^{\otimes r}\right),$$

and in characteristic zero define the log plurigenus by

$$p_g(X/s,r):={\rm \dim}_{\kappa}H^0\left(X,(\Omega^d_{X/s})^{\otimes r}\right).$$

Let $\overset{\circ}{X}$ denote the underlying scheme and $\overset{\circ}{X}{}^{(1)}$ its relevant base-change/Frobenius twist. Lower semi-continuity conjecture for log plurigenera. One should have

$$\begin{cases} p_g(\overset{\circ}{X}{}^{(1)}/\kappa,n,r)\leq p_g(X/s,n,r) & \text{if }\operatorname{ch}(\kappa)=p>0,\\ p_g(\overset{\circ}{X}{}^{(1)}/\kappa,r)\leq p_g(X/s,r) & \text{if }\operatorname{ch}(\kappa)=0. \end{cases}$$

This compares the plurigenera of the underlying scheme with the logarithmic plurigenera of the SNCL scheme. The paper explains that the finite-length argument does not prove the assertion for $r\geq2$, although the corresponding level-one positive-characteristic inequality and the characteristic-zero inequality are later established; the full stated level-$n$ claim therefore remains open in the source.