Balogh–Linz–Patkós' odd-parity conjecture for intersecting k-Sperner families

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote the Boolean lattice, and let a family $\mathcal{F}\subseteq 2^{[n]}$ be $t$-intersecting if $|A\cap B|\ge t$ for all $A,B\in\mathcal{F}$, and $k$-Sperner if it contains no chain of length $k+1$. When $n+t$ is odd, set

$$q=\frac{n+t-1}{2}.$$

For a fixed $t$-set $T\subseteq[n]$, define the $q$-uniform $t$-star and its upper shadow by

$$\mathcal{S}_T=\left\{S\in\binom{[n]}{q}:T\subseteq S\right\},$$ $$\nabla_s(\mathcal{F})=\left\{A\in\binom{[n]}{s}:\text{ there exists }F\in\mathcal{F}\text{ with }F\subseteq A\right\}$$

for $\mathcal{F}\subseteq\binom{[n]}{r}$ and $s\ge r$, and let

$$\mathcal{B}_T(t,k)=\mathcal{S}_T\cup\bigcup_{i=1}^{k-1}\binom{[n]}{q+i}\cup\left(\binom{[n]}{q+k}\setminus\nabla_{q+k}(\mathcal{S}_T)\right).$$

Its size is

$$|\mathcal{B}_T(t,k)|=\binom{n-t}{q-t}+\sum_{i=1}^{k}\binom{n}{q+i}-\binom{n-t}{q+k-t}.$$

Balogh–Linz–Patkós' conjecture. There exists a positive integer $n_0=n_0(k,t)$ such that if $n+t$ is odd, $n>n_0$, and $\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sperner family, then

$$|\mathcal{F}|\le\binom{n-t}{q-t}+\sum_{i=1}^{k}\binom{n}{q+i}-\binom{n-t}{q+k-t}.$$

This is the odd-parity counterpart of the even-parity extremal problem, with the fixed-star construction providing the predicted sharp bound. The source presents it as a conjecture; no resolution is supplied there.