The Haefliger-type h-principle conjecture for totally nonparallel immersions

Let $M$ be a smooth closed manifold of dimension $n$, and let $F_2(M)$ denote its ordered configuration space of two distinct points. The unordered configuration space is $F_2(M)/\mathbb Z_2$, where $\mathbb Z_2$ interchanges the two points. Let $q \geq \frac72 n$. Haefliger-type h-principle conjecture. If there exists an immersion

$$g:F_2(M)/\mathbb Z_2\to\mathbb R^q,$$

then there exists a totally nonparallel immersion $M\to\mathbb R^q$. This is proposed as a totally nonparallel analogue of Haefliger's embedding theorem: an immersion of the unordered configuration space should provide the obstruction-theoretic data needed to construct a totally nonparallel immersion in the indicated dimension range. The paper does not establish this converse, although it proves existence of totally nonparallel immersions into $\mathbb R^{4n-1}$ by other methods.