Threshold conjecture for positivity of partial sums of a random multiplicative function

Let $f$ be a random completely multiplicative function taking values in $\\{1,-1\\}$, with the values $f(p)$ chosen independently and uniformly at random over the primes. Let $\mathcal{L}_x^+$ be the set of such functions for which $\sum_{n\leq t}f(n)\geq 0$ for every integer $1\leq t\leq x$. For $y\geq 1$, condition on $f(p)=1$ for every prime $p\leq y$. Threshold conjecture. For every $\varepsilon>0$, there is $x_0(\varepsilon)$ such that, whenever $x>x_0(\varepsilon)$ and $y\leq(\log x)^{2-\varepsilon}$,

$$\operatorname{\mathbb{P}}\left(f\in\mathcal{L}_x^+\mid f(p)=1\\ (p\leq y)\right)=o(1).$$

Together with the proved upper-threshold result at $y\gg (\log x)^2\log_2x/\log_3x$, this would locate the transition for conditioned positivity near $y=(\log x)^{2+o(1)}$.