  In this article, we use the cone of nef curves to study minimal log
discrepancies. The first result is an improvement of the nef cone theorem in
the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain
condition for $n$-dimensional minimal log discrepancies of regularity one holds
around zero. Furthermore, we show that there exists an upper bound for the
minimal log discrepancy of any $n$-dimensional klt singularity of regularity
one.
