Concentration and separation conjecture for cycle lengths in Hamiltonian graphs

Let $G$ be an $n$-vertex Hamiltonian graph with minimum degree at least $3$. Concentration and separation conjecture. Both of the following hold:

1. $G$ has $\Omega(\sqrt n)$ cycle lengths all lying in an interval of size $O(\sqrt n)$.
2. $G$ has $\Omega(\sqrt n)$ cycle lengths such that any two are at least $\Omega(\sqrt n)$ apart.

These are proposed intermediate steps toward a linear lower bound on the number of cycle lengths. The source says that the main asymptotic conjecture implies this statement, but does not report a proof of either assertion, so it remains open.