Intersecting-family conjecture for Hamming powers of Kneser graphs

Let $B=\{0,1\}^n$, let $\mu$ be the uniform measure on $B^t$, and let $p_{\mathrm{intersecting}}(t)$ be the limiting maximal measure obtained when each player's winning family is an intersecting family, meaning that any two members share a coordinate equal to $1$. Intersecting-family conjecture. $p_{\mathrm{intersecting}}(t)$ tends to $0$ as $t$ grows. Since intersecting families correspond to independent sets in Hamming powers of Kneser graphs, this conjecture is a stronger graph-theoretic analogue of Levine's hats problem conjecture.