Do nonlinear Schrödinger terms built from antisymmetrized gradients vanish on all spin-coherent states with irrotational current?

A two-part question:

In the literature on experimental constraints on nonlinear modifications of quantum mechanics (Weinberg, Ann. Phys. 194, 336 (1989); Bollinger et al., PRL 63, 1031 (1989)), the tested nonlinearities are typically potential-type functionals of $|\psi|^2$. I've been looking at gradient-type quartics instead (they come about as small-amplitude limits of Skyrme-like commutator terms), and I've arrived at the following claim, which seems too strong, so I'd like to verify it. (It came out of AI-assisted work, which is why I'm looking for human verification.)

So, for a two-component spinor wavefunction $\psi$, define for Hermitian $\Gamma \in {1, \sigma^a}$:

$B^\Gamma_{ij}[\psi] = 2\,\mathrm{Im}\left(\partial_i \psi^\dagger\, \Gamma\, \partial_j \psi\right), \qquad E_{\rm NL}[\psi] = \sum_\Gamma g_\Gamma \int d^3x \sum_{i

Here is the claim: for any "spin-coherent" state $\psi = f(\mathbf{x})\chi$ with $\chi$ a constant spinor and $f = Re^{i\theta}$, you get

$B^\Gamma_{ij} = 2(\chi^\dagger \Gamma \chi)\, R\,(\partial_i R\, \partial_j \theta - \partial_j R\, \partial_i \theta),$

since $\chi^\dagger \Gamma \chi$ is real. So $E_{\rm NL}$ vanishes identically whenever the probability current $\mathbf{j} \propto R^2 \nabla\theta$ is irrotational, in particular for all real wavefunctions and all spherically spreading wave packets.

If this is correct, it would mean hydrogen 1S–2S spectroscopy, the Bollinger-type hyperfine tests, and wave-packet spreading experiments are all exactly blind to this operator class, and sensitivity requires spatial spin texture or orbital current (e.g., $m_\ell \neq 0$ or fine-structure-mixed states).

And here is the ywo-part question: (Pt. 1) Is the computation right, or am I missing a contribution? (Pt. 2) Is this class of nonlinearities, and its invisibility to these states, already treated somewhere in the constraints-on-nonlinear-QM literature?