Transversality conjecture for biharmonic almost complex structures

Let $(g,J)$ be a compatible almost Hermitian structure, and define

$$B(g,J)=\left[J,\,\Delta_g^2J\right].$$

A biharmonic almost complex structure is a zero of $B$. Transversality conjecture. The map $B$ is transverse to the zero section and hence the zero set of $B$ is regular. This would yield a standard moduli-space theory for biharmonic almost complex structures and, for generic metrics, metric-independent homotopy classes of energy-minimizing structures. The source does not provide a resolution of this transversality claim.