Conjectured formula for the first matching number of paired constructions

Let $n,k,t$ be non-negative integers with $k\leq n$ and $t\leq\min(k,n-k)$. Let $m_1(n,k,t)$ denote the first matching number associated with the paired construction for Boolean functions on the slice. The paired-construction matching-number conjecture.

$$m_1(n,k,t)= \begin{cases} 2\cdot\displaystyle\binom{2k-t-1}{k} & \text{for }n=2k\text{ and }t\text{ even and }t\neq k,\\[5mm] \displaystyle\binom{n-t}{\min(k,n-k)-t} & \text{otherwise.} \end{cases}$$

The conjecture gives a closed formula for the quantity $m_1(n,k,t)$ in all admissible parameter ranges, with a separate expression in the stated even case. The computational evidence preceding it does not establish the formula in general, so its resolution remains open.