Frankl's odd-parity extremal conjecture for t-intersecting k-Sperner families

Let $n,t,k$ be positive integers, and let ${\mathcal B}(t,k)$ denote the family constructed in the source: it contains the $(n+t-1)/2$-subsets containing $[1,t]$, all sets whose sizes range from $(n+t-1)/2+1$ through $(n+t-1)/2+k-1$, and the $(n+t-1)/2+k$-subsets not containing $[1,t]$. Let ${\mathcal F}\subseteq2^{[n]}$ be a $t$-intersecting family and a $k$-Sperner family. Frankl's conjecture. There exists a positive integer $n_0=n_0(k,t)$ such that, whenever $n+t$ is odd and $n>n_0$,

$$|{\mathcal F}|\leq|{\mathcal B}(t,k)|=\binom{n-t}{\frac{n-t-1}{2}}+\sum_{i=1}^{k}\binom{n}{\frac{n+t-1}{2}+i}-\binom{n-t}{\frac{n-t-1}{2}+k}.$$

The conjecture identifies ${\mathcal B}(t,k)$ as the largest such family for sufficiently large $n$; the source notes that a different candidate can be optimal for some small parameters.