The higher-jet transversality codimension conjecture

Let $X$ and $Y$ be finite-dimensional smooth manifolds, let $p\in\mathbb{N}$, and let $A\subseteq J^p(X,Y)$ be a smooth submanifold of codimension $c\leqslant n=\dim X$. Define

$$\tilde A=\big\{j^{p+1}_x f\in J^{p+1}(X,Y):j^p_xf\text{ is not transverse to }A\text{ at }x\big\}.$$

Higher-jet transversality codimension conjecture. The set $\tilde A$ is a countable union of smooth submanifolds of codimension more than $n=\dim X$ in $J^{p+1}(X,Y)$.

This conjecture would give the final case of the stated refinement of Thom's transversality theorem, by making the relevant exceptional set have codimension greater than the dimension of the source. The paper proves some cases but does not establish the full assertion.