Let $W$ be a one dimensional standard Brownian motion, and $\sigma: [0, \infty) \to \mathbb R$ a Borel function with $c < \sigma < C$ for some constants $c, C > 0$.

Does there exist some $M > 0$ such that, conditional on $W_t > -1$ for all $0 \leq t \leq 1$ and $W_1 > M$,

$\int_0^1 \sigma(t) \, dW_t \geq 0$, almost surely?

**Remark:** The statement would be true if the integrator had uniformly bounded variation, but I suspect it does not hold for Brownian motion which has unbounded variation.