Hamilton's Harnack-estimate conjecture for arbitrary initial metrics

Let $(M,g(t))$ be a solution to the Ricci flow on a compact $3$-manifold with arbitrary initial metric. Hamilton's Harnack-estimate conjecture. A Harnack-type estimate holds for this Ricci flow. Such an estimate would yield the Little Loop Lemma in the setting needed to exclude the cigar-soliton limits $\Sigma\times\mathbb{R}$ and $\Sigma\times S^1$, making it a crucial step in Hamilton's program toward Thurston's geometrization conjecture; the source gives no resolution.