Extended probability formula for piercing numbers in circular societies

Let $S$ be a random fixed-length circular society with $n$ voters, approval set length $p$, and piercing number $\tau(S)$. For a positive integer $k$, Extended probability formula. The probability that $S$ has piercing number $k$ is

$$P(\tau(S) = k) = \binom{n}{k} (1-kp)^{k-1}(kp)^{n-k}$$

for $p$ in an interval $(0, \frac{1}{2k} + \delta)$ for some $\delta > 0$. The formula is known for $p < \frac{1}{2k}$; the conjecture proposes that it continues to hold on a larger interval, as suggested by simulations for certain values of $k$.