Refinement of the biased r-wise intersection threshold conjecture

Let $n$ and $r$ be positive integers, let $[n]=\{1,2,\ldots,n\}$), and let $1>p_1\geq p_2\geq\cdots\geq p_n>0$. Write ${\bm p}=(p_1,p_2,\ldots,p_n)$. For a family $\mathcal A\subset 2^{[n]}$, let $\mu_{\bm p}(\mathcal A)$ denote its ${\bm p}$-biased measure, and call it $r$-wise intersecting if every $r$ members have nonempty total intersection.

Refinement of the biased $r$-wise intersection threshold conjecture. The condition

$$\frac{r-1}{r}>p_2$$

in the cited theorem can be replaced with

$$\frac{r-1}{r}>p_{r+1}.$$

In particular, the theorem holds if $p_4<\frac23$ instead of $p_3<\frac23$.

This is presented as a strengthening of a condition in an earlier theorem, rather than as an independently named result. The supplied text gives no resolution.