Smallest Dirac-eigenvalue contribution for even-dimensional spin Grassmannians

Let $G_{2k,2\ell}=SO(2(k+\ell))/(SO(2k)\times SO(2\ell))$ be the even-dimensional spin Grassmannian, and let $\lambda$ denote a weight occurring in the spinor-bundle decomposition. Smallest-contribution conjecture. The smallest contribution to the eigenspace corresponding to the smallest eigenvalue of the squared Dirac operator on $G_{2k,2\ell}$ is given by

$$\lambda^0=(\ell^{k-\ell+1},(\ell-1)^2,(\ell-2)^2,\ldots,1^2,0).$$

This is a more specific claim about the least contribution to the lowest Dirac eigenspace, following the displayed necessary norm condition; the parser supplies no evidence that it has been resolved.