For $i=1,2$, let $G_i=(A_i\cup B_i,E_i)$ be an undirected bipartite graph with bipartition $A_i$ and $B_i$, where $|A_1|=|A_2|=a$ and $|B_1|=|B_2|=b$ with $a\le b$.
Question. Is the problem of deciding isomorphism between such graphs solvable in $O(b^k 2^a)$ time, for some constant $k>0$?
Note that Babai's result yields a $2^{(\log(a+b))^c}$ time algorithm for some constant $c>1$. For the cases with $a=O(\log b)$, a solution to my question would yield a polynomial time complexity.