The degree-bounded SSM-Thom polynomial conjecture for Mather multisingularities

Assume $l\geq 1$. Let $\underline{\eta}$ be a Mather T- or S-multisingularity, let $M(l)$ be the Mather bound, and let $\mathcal{C}_l$ be the class of maps under consideration. Write $|_{\leq k}$ for projection to cohomological degrees at most $k$. Degree-bounded SSM-Thom polynomial conjecture. For the target case, for every degree bound $k\leq M(l)+l$ there exists a polynomial $\operatorname{Th}^T_{\underline{\eta},\leq k}\in\mathbb{Q}[\underline{s}]$ such that, for every $f:M\to N$ in $\mathcal{C}_l$,

$$\operatorname{Th}^T_{\underline{\eta},\leq k}(f)=\operatorname{ssm}(\Sigma^T_{\underline{\eta}}(f))_{|\leq k}\cdot|\operatorname{Aut}(\underline{\eta})|\in\operatorname{H}^{\leq k}(N).$$

For the source case, for every degree bound $k\leq M(l)$ there exists a polynomial $\operatorname{Th}^S_{\underline{\eta},\leq k}\in\mathbb{Q}[[\underline{c},\underline{s}]]$ such that

$$\operatorname{Th}^S_{\underline{\eta},\leq k}(f)=\operatorname{ssm}(\Sigma^S_{\underline{\eta}}(f))_{|\leq k}\cdot|\operatorname{Aut}(\underline{\eta})|\in\operatorname{H}^{\leq k}(M).$$

This is the degree-cut version of the SSM-Thom principle in the Mather nice dimension range. The source reports that it can be verified in many cases, but does not establish it in full generality.