Define $L(x) = \sum_{n\leq x} \lambda(n)$, where $\lambda$ denotes the Lioville function. If $c$ is the supremum of the real parts of the zeros of the Riemann zeta function and $[x]$ denotes the integer part of $x$, is it true that

$$\Big|\int_{1}^{x} L(y) [x/y] \frac{\mathrm{d}y}{y} \Big| > x^{c-\epsilon}$$ for arbitrarily large $x$ and any $\epsilon>0$ ?