Conjecture on arbitrarily oscillating win-probability derivatives

Let ${\normalfont{\operatorname{Win}}}(v,w;p)$ denote the probability that the binary word $v$ occurs before the binary word $w$ in an iid Bernoulli$(p)$ sequence.

Oscillation conjecture. There exists a constant $C>0$ such that, for every positive integer $k$, there is a pair of binary words $(v,w)$ with lengths at most $\exp(Ck)$ for which

$$\frac{d}{dp}{\normalfont{\operatorname{Win}}}(v,w;p)$$

changes sign at least $k$ times as $p$ ranges over $(0,1)$.

Win probabilities are rational functions of $p$, with endpoint values constrained to $\{0,\frac12,1\}$. The conjecture predicts that despite this algebraic structure, derivatives of win-probability functions can exhibit arbitrarily many sign changes using pairs whose word lengths grow only exponentially in the number of changes. The source presents this as supported by heuristic arguments and computer checks, with no resolution given.