= Solution
Assume the scalar potential is real, so the <Weyl equation> has a Hermitian Hamiltonian $H=\boldsymbol\sigma\cdot\mathbf p+V$. Multiplying the equation and its adjoint by $\psi^\dagger$ and $\psi$ respectively gives
$$
\partial_t(\psi^\dagger\psi)=-\boldsymbol\nabla\cdot(\psi^\dagger\boldsymbol\sigma\psi).
$$
Thus with $\rho=\psi^\dagger\psi$ and $\mathbf J=\psi^\dagger\boldsymbol\sigma\psi$, the lower-index four-current $j_\mu=(\rho,-\mathbf J)$ obeys $\boxed{\partial^\mu j_\mu=0}$.
For $K=\boldsymbol\sigma\cdot\mathbf p$,
$$
[K,H]=[K,V]=-i\boldsymbol\sigma\cdot\boldsymbol\nabla V.
$$
Consequently $K$ is conserved for spatially constant $V$; a spatially uniform time-dependent scalar offset would also commute with it. For general nonuniform $V$ it is not conserved. At nonzero momentum, $(\boldsymbol\sigma\cdot\hat{\mathbf p})^2=I$, so the <helicity> eigenvalues are $\boxed{+1,-1}$. In the chosen constant-potential Weyl Hamiltonian, the positive $E-V$ branch has helicity $+1$; the negative branch has $-1$. An expectation value in a superposition need not itself be an eigenvalue.
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