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

Let $n$ and $t$ be integers with $n+t$ even and $n>t$, and let ${\mathcal F}\subseteq 2^{[n]}$ be a $t$-intersecting family, meaning that $|F\cap F'|\geq t$ for all $F,F'\in{\mathcal F}$, and a $k$-Sperner family, meaning that it contains no chain of $k+1$ distinct sets under inclusion. Frankl's conjecture. One has

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

The bound is tight if true, attained by $\bigcup_{i=0}^{k-1}\binom{[n]}{\frac{n+t}{2}+i}$.