Prove that the number of nonisomorphic ways to partition $E(k_n)$ into complete bipartite graphs at least $2^{n-4}$

Question

Apparently the solution is exercise 1.4.5 in Babal and Frankl's "linear algebra methods in combinatorics (1992)" but it seems they never actually got around to publishing this. Or at least, I am unable to aquire it.

Any advice appreciated. I think maybe the solution relies on the odd/even town theorem but I'm not too sure.

Thanks!