Rudyak's conjecture for distributional category

Let $M$ and $N$ be closed orientable $n$-manifolds, and let $f:M\to N$ be a map of degree $\pm1$. Write $\operatorname{\mathsf{dcat}}$ for the distributional category of a manifold. Rudyak's conjecture. If $f$ has degree $\pm1$, then

$$\operatorname{\mathsf{dcat}}(M)\ge \operatorname{\mathsf{dcat}}(N).$$

This is the distributional-category analogue of Rudyak's conjecture for the classical Lusternik–Schnirelmann category, which asserts the corresponding inequality for $\operatorname{\mathsf{cat}}$. The source presents the statement as a conjecture; its resolution is not specified here.