Near-perfect clique-factors in sparse pseudorandom graphs

We prove that, for any $t\ge 3$, there exists a constant $c=c\left(t\right)>0$ such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d}^{t-1}/n^{t-2}$ contains $(1-o(1\left)\right)n/t$ vertex-disjoint copies of $K_t$. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), pp. 403–426] that ($n,d,\lambda$)-graphs with $n\in 3N$ and $\lambda \le c{d}^2$ for a suitably small absolute constant $c>0$ contain triangle-factors.