Let $F(s)$ be a Titchmarsh series, i.e. a kind of Dirichlet series, the coefficients of which are suitably bounded. This notion was introduced by us in an earlier paper where we stated a conjecture and proved certain theorems on the lower bound for $\int_0^H \vert F(it)\vert^k\,dt$, where $k=1$ or $2$. In this paper, better results and a proof of the conjecture are obtained.