Let F be a field, β a prime and D a central division F -algebra of β-power degree. By the Rost kernel of D we mean the subgroup of F * consisting of elements Ξ» such that the cohomology class (D) βͺ (Ξ») β H 3 (F, Q β /Z β (2)) vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by i-th powers of reduced norms from D βi , βi β₯ 1. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume F is a henselian discrete valuation field with residue field k of characteristic different from β. When D has period β, we show that Suslin's conjecture holds if either k is a 2-local field or the cohomological β-dimension cd β (k) of k is β€ 2. When the period is arbitrary, we prove the same result when k itself is a henselian discrete valuation field with cd β (k) β€ 2. In the case β = char(k) an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.