Asymptotic rarity of high chromatic index in Steiner triple systems

Let $m(v)$ be the lower bound for the chromatic index of a Steiner triple system of order $v$, namely

$$m(v)=\begin{cases}\frac{v-1}{2}&\text{if }v\equiv3\pmod 6,\\[3pt]\frac{v+1}{2}&\text{if }v\equiv1\pmod 6. \end{cases}$$

Asymptotic rarity conjecture. The proportion of Steiner triple systems of order $v$ whose chromatic index is at least $m(v)+2$ tends to $0$ as $v\to\infty$.

This conjecture is motivated by computational observations in the source: systems with chromatic index $m(v)+2$ were rare in the small orders examined, although the experiments were not sampled from a uniform distribution. The source gives no resolution.