A graph G is K_r-covered if each vertex of G is contained in a K_r-clique. Let γ_t(G) denote the total domination number of G. It has been conjectured that every K_r-covered graph of order n with no K_r-component satisfies γ_t(G) ≤ 2n/(r+1). We prove that this conjecture is true for r=5 and 6.