Frankl's conjecture for even $n+t$ and $t$-intersecting $k$-Sperner families

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq 2^{[n]}$. The family $\mathcal{F}$ is $t$-intersecting if $|A\cap B|\geq t$ for all $A,B\in\mathcal{F}$, and it is $k$-Sperner if it contains no chain of length $k+1$. Suppose that $n+t$ is even and $n>t$. Frankl's conjecture. If $\mathcal{F}$ is a $t$-intersecting $k$-Sperner family, then

$$|\mathcal{F}|\leq \sum_{i=0}^{k-1}\binom{n}{(n+t)/2+i}.$$

The conjectured extremal construction consists of the $k$ consecutive layers beginning at level $(n+t)/2$. Frankl proved the conjecture in several ranges; the general common extension of Milner's and Frankl's theorems remains open.