DP-completeness of recognizing minimally $t$-tough graphs

Let $t$ be a positive rational number. The decision problem ${\scshape Min-}t$-Tough asks whether a graph is minimally $t$-tough, meaning that its toughness is at least $t$ and deleting any edge decreases its toughness below $t$.

Min-$t$-Tough complexity conjecture. ${\scshape Min-}t$-Tough is DP-complete for any positive rational number $t$.

The paper proves DP-completeness for some positive rational values of $t$, while the remaining values are stated to be open; thus the assertion is unresolved in the full stated range.