The balanced-monotone-family success-probability conjecture

Let $B=\{0,1\}^n$, and let $\mu$ be the uniform measure on $B^t$. A balanced monotone family is a set $W\subseteq\{0,1\}^n$ containing precisely half the points and such that $x\in W$ and $y_i\geq x_i$ for all $i$ imply $y\in W$. Let $p_{\mathrm{monotone}}(t,n)$ be the corresponding maximal success probability, and let

$$p_{\mathrm{monotone}}(t):=\lim_{n\to\infty}p_{\mathrm{monotone}}(t,n).$$

Balanced-monotone-family conjecture. The success probability $p_{\mathrm{monotone}}(t)$ tends to $0$ as $t$ grows. This is stronger than the intersecting-family conjecture because maximal intersecting families are balanced monotone families; its status is open in the source.