Joseph's conjectures on limit points of commuting probabilities

Let $S=\{P(G):G\text{ is a finite group}\}\subseteq(0,1]$, where $P(G)$ is the commuting probability of $G$. Consider a sequence $(x_i)_{i=1}^\infty$ of elements of $S$ converging to $\ell>0$.

Joseph's conjectures. One should have $\ell\in\mathbb{Q}$, $x_i\geq\ell$ for all but finitely many $i$, and $\ell\in S$. Equivalently, the second conjecture says that $S$ is well-ordered with respect to the opposite ordering, and the third says that $S\cup\{0\}$ is closed.

The conjectures describe the limiting behavior of commuting probabilities of finite groups. The third conjecture, that $S\cup\{0\}$ is closed, is proved in this paper; consequently, the combined conjectural package is not entirely open.