Probabilistic upper-bound conjecture for the maximum planar EMST-ratio

Let $P_n$ consist of $n$ independent random points uniformly distributed in $[0,1]^2$. Probabilistic upper-bound conjecture. The maximum EMST-ratio of $P_n$ is less than $2$ with probability tending to $1$ as $n\to\infty$.

The conjecture is motivated by experiments on random point sets, which found no maximum EMST-ratio greater than $2$ for the tested sizes. The source separately asks for the limiting expected maximum EMST-ratio.