Fold-lift conjecture for Morin maps

Let $f:M\to N$ be a Morin map. A lift of $f$ is a map $\tilde{f}:M\to N\times\mathbb{R}$ whose composition with the projection $N\times\mathbb{R}\to N$ is $f$. Fold-lift conjecture. There is a lift $\tilde{f}:M\to N\times\mathbb{R}$ of $f$ which is a fold map. This is an obstruction-theoretic formulation concerning whether Morin maps can be lifted to fold maps; the extracted text gives no resolution status beyond the conjectural formulation.