McDiarmid and Scott's conjecture on the diameter of block-stable random graphs

Let $\mathsf{C}_n$ be a uniform random graph from a block class, with each block receiving weight $0$ or $1$, and let $t_n$ be a sequence tending to infinity. McDiarmid and Scott's conjecture. With high probability, every path in $\mathsf{C}_n$ passes through at most $t_n\sqrt{n}$ blocks. This conjecture predicts that the previously proved upper bound $5\sqrt{n\log(n)}$ can be improved by replacing the factor $\sqrt{\log(n)}$ with any sequence tending to infinity.