Equivariant Chern class formula for arc spaces of the general linear group

Let $G=GL_n$ and let $V_\bullet$ be the universal flag representation used to define the contact loci $\operatorname{Cont}^{\geq\lambda}(V_\bullet)$ and $\operatorname{Cont}^{\lambda}(V_\bullet)$. Recall that $\Lambda_G=\mathbb{Z}[c_1,\ldots,c_n]$, where $c_i$ is the $i$th equivariant Chern class of the standard representation. For a tuple of non-negative integers $\mathbf{m}=(m_1,\ldots,m_n)$, let $\lambda=\lambda(\mathbf{m})$. Equivariant Chern class formula. One has

$$[\operatorname{Cont}^{\geq \lambda}(V_\bullet)]=[\overline{\operatorname{Cont}^{\lambda}(V_\bullet)}]=c_1^{m_1}\cdots c_n^{m_n}.$$

The preceding corollary shows that these orbit-closure classes form a $\mathbb{Z}$-basis of $\Lambda_G$; the conjecture identifies this geometric basis with the monomial basis in equivariant Chern classes. Its status is not resolved in the supplied text.