Cobordism conjecture for Morin and de-suspended singular loci of twisted prim maps

Let $f\colon M^n\to N^{n+k}$ be a twisted prim map, where $M$ is closed and $k\geq r-1$, and let $\tilde f\colon M\times\mathbb{R}^{r-1}\to N$ be a de-suspension of $f$. The submanifolds $\overline{\Sigma}^{1_r}(f)$ and $\Sigma^r(\tilde f)\cap (M\times\{0\})$ of $M$ are embedded-cobordant in $M$.

Cobordism conjecture. This cobordism is oriented if $M$ and $N$ are oriented, the codimension $k$ is odd, and $r$ is even.

The conjecture would remove the factors $2$ in the corresponding theorem and corollary for twisted prim maps, strengthening the known coincidence of the associated Thom-polynomial classes to a geometric equivalence of the singular loci. The source presents this as an open conjecture.