First of all, I am confused about the WZW model itself. Let us consider the following WZW-NLSM model on 2-D manifold $X_2=\partial(X_3)$ with level-1 $SO(4)$ WZW term: \begin{eqnarray} S=\int_{X_2}\frac{(\partial\phi)^2}{2g^2}+\frac{i}{\pi}\int_{X_3}\frac{\epsilon_{ijkl}}{3!}\phi^id\phi^jd\phi^kd\phi^l \end{eqnarray} where $\phi$ is a four-component vector in the fundamental $SO(4)$ representation.
I am reading one paper "Competing Orders and Anomalies" https://arxiv.org/abs/1503.05199 which gives an explanation to the fact that, although the WZ term is defined on $X_3$, its equation of motion is still on $X_2$. It states that this fact is resultant from the closeness of WZ term on $X_3$. At the first glance, it indeed makes sense to me. However, that paper uses the similar explanation to deal with the gauged WZW model where $SO(4)$ is gauged by a connection $A$: \begin{eqnarray} \tilde{S}&=&\int_{X_2}\frac{(D\phi)^2}{2g^2}+\frac{i}{\pi}\int_{X_3}\frac{\epsilon_{ijkl}}{3!}\phi^iD\phi^jD\phi^kD\phi^l,\\ D&\equiv&d+A. \end{eqnarray} However, the author somehow argued that the gauged WZ term in $\tilde{S}$ defined above is not closed (thereby needing one more Chern-Simons term)! But by a simple counting, $\phi^iD\phi^jD\phi^kD\phi^l$ is clearly still the highest form on $X_3$, which must be closed. Thus I wonder whether such naive counting is wrong, or whether there exists other obstruction for a highest form to be closed?
Or whether there exists other nice explanation or understanding about the above problem? I am a student in Physics and really appreciate it if you could give me a more math-friendly answer.
Thanks very much in advance!