We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-α})_{k=1}^\infty$ and centres chosen independently at random according to an arbitrary Borel probability measure $μ$ on $\mathbb{R}^d$. The conjecture has been solved positively in the case $\frac 1α\le \overline{\dim}_H μ$, where $\overline{\dim}_H μ$ stands for the upper Hausdorff dimension of $μ$. In this paper, we develop a new approach in order to answer the full conjecture, proving in particular that the conjectured value is only a lower bound for the dimension. Our approach opens the way to study more general limsup sets, and has consequences on the so-called hitting probability questions. For instance, we are able to determine whether and what part of a deterministic analytic set can be hit by random covering sets formed by open sets.