On the minimum number of triangles in balanced tripartite graphs with large minimum degree

Let $f(n,t)$ be the minimum number of triangles in a tripartite graph with $n$ vertices in each part and minimum degree at least $n+t$. In 1975, Bollobás, Erdős and Szemerédi proved that $f(n,1)=\min\{4,n\}$. They further remarked that it is ``very likely'' that $f(n,t)\ge4t^3$ for $n\ge5t$. They al