Conjecture on powers of Hamilton cycles in pseudorandom graphs

For $k\geq 2$, let an $(\varepsilon,p,k-1,k)$-pseudorandom graph mean a graph satisfying the stated pseudorandomness condition with parameters $(\varepsilon,p,k-1,k)$. The $k$th power of a Hamilton cycle is obtained by joining every pair of vertices whose distance on the cycle is at most $k$. Powers-of-Hamilton-cycles conjecture. For all $k\geq 2$, the pseudorandomness requirement in Theorem 1 can be replaced by $(\varepsilon,p,k-1,k)$-pseudorandomness. This would extend the result proved in the paper for squared Hamilton cycles to all higher powers, under a weaker pseudorandomness requirement.