For any $k$, the class of graphs of treewidth at most $k$ can be characterized by a finite set of forbidden minors.

For treewidth $1$ and $2$, the set is of size $1$. Then for treewidth $3$, the set is of size $4$, and beyond that the size of the set grows very rapidly.

For a given $k$, is there a known upper bound on the number of vertices for the forbidden minors for treewidth at most $k$?