C. C. Lindner and T. Evans [Finite embedding theorems for partial designs and algebras (Séminaire de Mathématiques Supérieures 56, Les Presses de l’Université de Montréal) (1977; Zbl 0363.05017)] conjectured that any partial Steiner triple system of order \(u\) can be embedded in a Steiner triple system of order \(v\) if \(v \geq 2u+1\) and \(v \equiv 1,3 {\pmod 6}\). To date this conjecture remains unresolved. In this paper the author presents a conjecture that implies the above conjecture, and which also has implications for embeddings of order \(v < 2u+1\). A partial Steiner triple system of order \(u\) with leave \(L\) can be embedded in a Steiner triple system of order \(v = u+w\) if and only if there is a \(K_3\)-decomposition of \(L \vee K_w\). The author proves a lemma that establishes necessary conditions for the existence of such \(K_3\)-decompositions. His conjecture is that these necessary conditions are also sufficient. He backs this up by proving several supporting results, including the fact that the conjecture holds for graphs \(L\) with maximum degree at most \(2\) in the case \(w < u+1\). This result yields necessary and sufficient conditions for embeddings of partial Steiner triple systems of order \(u\) with leaves having maximum degree 2 in Steiner triple systems of order \(v < 2u+1\).