Stable-envelope classes for the attracting stratification

Let $\mathfrak M_1$ be the variety with finite fixed-point set $\mathfrak M_1^T$, let $\mathfrak Z_p$ denote the attracting set of $p\in\mathfrak M_1^T$, and let $\mathfrak Z$ be the attracting scheme. Write $\operatorname{cf}_p(\gamma)$ for the coefficient of a class $\gamma$ at the fixed point $p$ under equivariant localization, and let $\varphi$ be the equivariant parameter appearing in the specialization condition.

Stable-envelope existence and uniqueness conjecture. There exist unique classes $(S_p)_{p\in\mathfrak M_1^T}$ in $H_{\dim\mathfrak Z}^T(\mathfrak M_1)$ such that, for every $p\in\mathfrak M_1^T$:

1. The support of $S_p$ lies inside $\bigsqcup_{q\leq p}\mathfrak Z_q$, meaning that a class on this union pushes forward to $S_p$.
2. $\operatorname{cf}_p(S_p)=\operatorname{cf}_p(\overline{\mathfrak Z_p})$.
3. $\operatorname{cf}_q(S_p)|_{\varphi=0}=0$ for $q<p$.

These conditions reformulate the stable envelope construction in homological language. The supplied text does not provide evidence resolving whether the stated classes exist uniquely, so the conjecture remains open.