Rudyak's conjecture for lower dimensional 1-connected manifolds

Rudyak’s conjecture states that for any degree one map $f:M\to N$ between oriented closed manifolds there is the inequality $\text{cat}(M)\ge \text{cat}(N)$ for the Lusternik-Shnirelmann category. We prove the Rudyak’s conjecture for $n$-dimensional simply connected spin manifolds for $n\le 8$.