  Katona and Varga showed that for any rational number $t \in (1/2,1]$, no
chordal graph is minimally $t$-tough. We conjecture that no chordal graph is
minimally $t$-tough for $t>1/2$ and prove several results supporting the
conjecture. In particular, we show that for $t>1/2$, no strongly chordal graph
is minimally $t$-tough, no split graph is minimally $t$-tough, and no chordal
graph with a universal vertex is minimally $t$-tough.
