Equivariant Chern class formula for flag varieties via arc spaces

Let $G=GL_n$, let $\mathbf{r}=(r_1,\ldots,r_k)$ determine the partial flag variety and the corresponding flag representation $\widetilde{V}_\bullet$, and let $\widetilde{\lambda}=\widetilde{\lambda}(\mathbf{m})$ for a tuple of non-negative integers $\mathbf{m}=(m_1,\ldots,m_n)$. The classes $[\overline{\operatorname{Cont}^{\widetilde{\lambda}}(\widetilde{V}_\bullet)}]$ form a $\mathbb{Z}$-basis of $H_G^*Fl(\mathbf{r})=\Lambda_P$, where $c_{j,\mathbf{r}}$ denotes the specified elementary symmetric function in the relevant block of variables. Equivariant flag-variety formula. One has

$$[\operatorname{Cont}^{\geq \widetilde{\lambda}}(\widetilde{V}_\bullet)]=[\overline{\operatorname{Cont}^{\widetilde{\lambda}}(\widetilde{V}_\bullet)}]=(-1)^{\sum_i\widetilde{\lambda}_i}c_{1,\mathbf{r}}^{m_1}\cdots c_{n,\mathbf{r}}^{m_n}.$$

The conjecture would identify the geometric basis of the equivariant cohomology of the partial flag variety with the indicated multiply-symmetric-function monomials, including the stated sign. Its status is not resolved in the supplied text.