Poisson limit conjecture for subsystems of Steiner triple systems

Let $S$ be the set of positive integers for which Steiner triple systems of order $v$ exist. For an Steiner triple system of order $v$, let the number of its subsystems of order $w$ be the random variable under the uniform distribution on Steiner triple systems of order $v$. Poisson limit conjecture. As $v\to\fty$ through values in $S$, the distribution of the number of subsystems of order $w$ tends to a Poisson distribution with expected value $1/168$ when $w=7$ and expected value $0$ when $w>7$. The conjecture is motivated by a random hypergraph model whose limiting expected values are $1/168$ for order-seven subsystems and $0$ for larger subsystems; the source provides no proof or resolution.