Conjecture on totally chain-intersecting families for p at least q

Let $[n]=\{1,\ldots,n\}$ and let $\mathcal F\subseteq 2^{[n]}$ be a family. For positive integers $p$ and $q$, call $\mathcal F$ totally $(p,q)$-chain intersecting if it contains no chains $A_1\subsetneq\cdots\subsetneq A_p$ and $B_1\subsetneq\cdots\subsetneq B_q$ with $A_1\cap B_1=\emptyset$. For $1\leq i\leq n$, define

$$\mathcal F_q(i)=\{F\subseteq 2^{[n]}:1\in F,\ |F|\leq i\}\cup\{F\subseteq 2^{[n]}:i-q+1\leq |F|\leq i\}.$$

Let $\mathcal R$ denote the middle $p-1$ levels.

Conjecture for totally chain-intersecting families. If $\mathcal F$ is a totally $(p,q)$-chain intersecting family and $p\geq q$, then

$$|\mathcal F|\leq\max\{ |\mathcal F_q(i)|,|\mathcal R|\}.$$

The source explains that $\mathcal F_q(i)$ is totally $(p,q)$-chain intersecting when $p\geq q$, providing the proposed extremal construction. No resolution of this conjecture is stated in the supplied text.