Define $L(x) = \sum_{n\leq x} \lambda(n)$, where $\lambda$ denotes the [Liouville function](https://en.wikipedia.org/wiki/Liouville_function). If $c$ is the supremum of the real parts of the zeros of the Riemann zeta function and $\lfloor x\rfloor$ denotes the integer part of $x$, is it true that 
$$\Bigl\lvert\int_{1}^{x} L(y)\lfloor x/y\rfloor\frac{\mathrm{d}y}{y} \Bigr\rvert > x^{c-\epsilon}$$ for arbitrarily large $x$ and any $\epsilon>0$?