Conjecture on optimal stability constants for 3-wise intersecting families

Let $0<p<1$, let $n$ be a positive integer, and let ${\bm p}=(p,p,\ldots,p)\in(0,1)^n$. Let $\mathcal A$ be a 3-wise intersecting family, and let $\mathcal B$ be the corresponding star from the stability statement in the cited theorem. Write $\mathcal A\triangle\mathcal B$ for the symmetric difference, and let $\epsilon$ denote the parameter in that stability theorem. Let $C_p$ be the constant appearing there.

Conjecture on optimal stability constants. There exists a constant $C_p'$ such that the inequality in case (ii) of the theorem can be replaced by

$$\mu_{\bm p}(\mathcal A\triangle\mathcal B)<(C_p'+o(1))\epsilon,$$

where $C_p'\leq C_p$ and $C_p'$ is increasing in $p$ for $\frac12\leq p<\frac23$.

The conjecture seeks a sharper and more regular stability constant near $p=\frac12$. The supplied text gives no resolution.