The Prem Conjecture for smoothly k-realizable maps

Let $N^n$ be a smooth manifold and let $f:N^n\to\mathbb{R}^m$ be a smoothly $k$-realizable general-position smooth map, meaning that the composition

$$N\xrightarrow{f}\mathbb{R}^m\subset\mathbb{R}^m\times\mathbb{R}^k$$

is $C^0$-approximable by smooth embeddings. A smooth map is a smooth $k$-prem if it admits a smooth lifting to an embedding into $\mathbb{R}^m\times\mathbb{R}^k$ whose projection to $\mathbb{R}^m$ is the given map.

The Prem Conjecture. Smoothly $k$-realizable general-position smooth maps $N^n\to\mathbb{R}^m$ are smooth $k$-prems, at least in the metastable range

$$2(m+k)\geq 3(n+1).$$

The conjecture relates approximation of the graph of a map by embeddings to the existence of an actual projected embedding. In the stated generality it has been resolved: every $1$-realizable stable smooth map $N^n\to\mathbb{R}^{2n-1}$, for $n\geq 3$, is a smooth $1$-prem.