Tightness conjecture for the $(n,n,\binom{n}{2})$ balanced matching bound

Let $\textsc{bm}(a_1,a_2,t)$ denote the balanced matching parameter for the indicated complete bipartite setting. Tightness conjecture. For every integer $n\geq4$,

$$\textsc{bm}\left(n,n,\binom{n}{2}+1\right)\xrightarrow{~\tiny \textsc{bm}~}n.$$

This would make the preceding upper bound $\textsc{bm}(n,n,\binom{n}{2})\leq n-1$ tight in the stated sense; the source presents it as a conjecture and gives no resolution.