The conjectured optimal bound for completely multiplicative functions with mean value zero

Let $x$ be large, and let $f$ be a completely multiplicative function satisfying $-1\leqslant f(n)\leqslant 1$ for all $n$. Suppose that

$$\sum_{n\leqslant x}f(n)=o(x).$$

Conjectured optimal bound. For $1/\sqrt e\leqslant \alpha\leqslant 1$, one has

$$\left|\sum_{n\leqslant x^{\alpha}}f(n)\right|\leqslant \bigl(-2\log\alpha+o(1)\bigr)x^{\alpha}.$$

This would give a sharp form of the preceding Burgess-type estimate for partial sums of completely multiplicative functions whose average up to $x$ is $o(1)$. The supplied text does not state whether the conjecture has since been proved or disproved.