A reconstruction claim for binomial partial Steiner triple systems

Let $m<n$ and let $\left({\goth N}_i\colon i=1,\ldots,m\right)$ be a sequence of $\binkonf(n,0)$-configurations. The source proposes that each $\goth N_i$ freely contains $m-1$ graphs $K_{n-1}$, with an additional compatibility condition on their complementary subconfigurations, although that condition is not specified.

Reconstruction claim. There are two nonisomorphic $\binkonf(n,1)$-configurations, each with $m$ complete subgraphs $K_n$, such that the corresponding complementary subconfigurations are the prescribed $\goth N_i$.

This appears in a classification discussion about reconstructing binomial $\binkonf(n,1)$-configurations from complementary $\binkonf(n,0)$-configurations. Because the second hypothesis is incomplete and the parser marks the status as unknown, the precise mathematical claim and its resolution require verification against the source.