Higher Wess-Zumino-Witten term from semistrict higher Chern-Simons theory

We show that in any $2n+2$ dimensions, the higher Chern-Simons action built from a semistrict Lie 2-algebra gives a non-trivial higher Wess-Zumino-Witten (WZW) term under a higher gauge transformation. The key point is that the non-zero Jacobiator directly generates the higher WZW term, hereas it vanishes when reduced to the strict case.