Monotonicity conjecture for favorable longer-word pairs

Let $n$ be a sufficiently large positive integer. For binary words $v,w$ of length at most $n$, let $\tau_v$ and $\tau_w$ be their first occurrence times in an iid Bernoulli$(p)$ sequence, and let $v$ be longer than $w$.

Monotonicity conjecture. The number of pairs $(v,w)$ satisfying

$$\mathbb{P}_p(\tau_v < \tau_w)>\frac{1}{2}$$

is non-increasing as a function of $p\in\left(0,\frac{1}{2}\right)$.

This conjecture formalizes the expected increase, as the Bernoulli parameter decreases, in the frequency with which a longer word defeats a shorter word. It is supported by computer computations but is disproved by the example $(v,w)=(100010,001100)$, whose win probability is $\frac{1}{2}(1-p(1-p)^4)$ and is minimized at $p=\frac{1}{5}$, so the asserted monotonicity fails.