Higher-dimensional Logarithmic Extension Theorem conjecture for log canonical pairs

Let $k$ be a perfect field of characteristic $p\geq 7$, and let $(X,\emptyset)$ be a log canonical pair over $k$ of dimension at most $p-2$. Logarithmic Extension Theorem conjecture. The Logarithmic Extension Theorem holds for $(X,\emptyset)$. The conjecture concerns the range not excluded by the higher-dimensional counterexamples: in characteristic $p$, such counterexamples begin in dimension $p-1$, while the assertion remains open in dimensions at most $p-2$.