Buzzard–Gee irreducibility conjecture with explicit exceptional conditions

Let $r$ be a positive integer written as $r=t(p-1)+s$ with $t\geqslant 0$ and $s\in\{1,\ldots,p-1\}$, let $a\in\overline{\mathbb Q}_p$ with $v=v(a)\geqslant 1$, and define the rising factorial by

$$x^{(n)}=x(x+1)\cdots(x+n-1).$$

Let $\overline V_{r+2,a}$ denote the semisimplified reduction of the associated crystalline representation. Buzzard–Gee conjecture. If $r$ is even and $v\notin\mathbb Z$, then $\overline V_{r+2,a}$ is irreducible in the following cases: (1) $s\in\{2,\ldots,2\lfloor v\rfloor\}$ and $p\nmid (r-s)^{(2\lfloor v\rfloor+1)}$; (2) $s\notin\{2,\ldots,2\lfloor v\rfloor\}$ and $p\nmid (r-s)^{(\lfloor v\rfloor)}$, in which case

$$\overline V_{r+2,a}\cong\operatorname{ind}(\omega_2^{s+(p-1)\lfloor v\rfloor+1});$$

or (3) $s\notin\{2,\ldots,2\lfloor v\rfloor\}$. The paper proves the three parts for, respectively, $v<6$, $v<38$, and $v<3$, so the remaining ranges are open.