The generic coincidence conjecture for approximated partial limits

Let $\left\{a_n^{(\omega)}\right\}_{n=1}^{\infty}$ range over all possible sequences, and let $p\tilde{\lim}a_n^{(\omega)}$ denote the set of approximated partial limits while the partial limits have their usual meaning. Generic coincidence conjecture. The set of all sequences $\left\{\left\{a_n^{(\omega)}\right\}_{n=1}^{\infty}\right\}_{\omega}$ for which

$$p\tilde{\lim}a_n^{(\omega)}\cap\{\text{partial limits of }\left\{a_n^{(\omega)}\right\}_{n=1}^{\infty}\}=\varnothing$$

is negligible with respect to the set of all possible sequences. The meaning of “negligible” and the measure or genericity structure on the space of sequences are not specified in the supplied text, so the conjecture's precise interpretation and status require verification.