More on halving the complete designs (Q1343246)

\textit{Alan Hartman} [Ann. Discrete Math. 34, 207-224 (1987; Zbl 0643.05013)] conjectured that the complete \(S((\begin{smallmatrix} v-t\\ k- t\end{smallmatrix}); t,k,v)\) can be partitioned into two \(S((\begin{smallmatrix} v-t\\ k-t\end{smallmatrix})/2; t,k,v)\) if and only if \((\begin{smallmatrix} v- i\\ k- i\end{smallmatrix})\) is even for all \(i= 0,\dots, t\). This conjecture remains open although several partial results are known. (A review of these results is given in the paper.) In the present paper, the authors prove that the conjecture is true for \(t= 2\) and \(2\leq k\leq 15\).