Cycle-count conjecture for degree 3-critical graphs

A degree 3-critical graph is a graph on $n$ vertices with $2n-2$ edges and no proper induced subgraph of minimum degree $3$. Cycle-count conjecture. Every degree $3$-critical graph on $n$ vertices contains cycles of at least

$$3\log_2 n+O(1)$$

distinct lengths. The conjecture is motivated by the Bollobás–Brightwell construction, which gives degree $3$-critical graphs with no cycles longer than $4\log_2 n+O(1)$. Determining the minimum possible number of distinct cycle lengths remains open.