Smallest-eigenvalue weights for even-dimensional spin Grassmannians

Let $G_{2k,2\ell}=SO(2k+2\ell)/(SO(2k)\times SO(2\ell))$ be the even-dimensional spin Grassmannian, and let the spinor-bundle decomposition be the one described in the preceding decomposition claim. Suppose $\ell\leq k$. Smallest-eigenvalue weight conjecture. Exactly $2^\ell$ weights in the spinor-bundle decomposition are associated with the smallest eigenvalue of the square of the Dirac operator; they are

$$(\ell^{k-\lambda_1},(\ell-1)^{\lambda_1-\lambda_2},\ldots,1^{\lambda_{\ell-1}-\lambda_\ell},0^{\lambda_\ell}\mid\lambda_1,\ldots,\lambda_\ell),$$

where $\lambda_j\in\{\ell-j+1,\ell-j\}$. This conjecture identifies the full family of weights contributing to the lowest Dirac eigenspace, complementing the preceding proposition for the value of the smallest eigenvalue; no resolution is supplied.