Conjectured parameter ranges for two Gaussian hypergeometric inequalities

Let $a,b>0$ and $r\in(0,1)$. Let $\Lambda(a,r)$ and $\Lambda_2(a,b_1,b_2,r)$ denote the quantities defined earlier in the source by the inequalities referenced as

$$and$$

, respectively. Parameter-range conjecture. The inequality

holds for every $r\in(0,1)$ and every $a>0$ if and only if

b\in\bigl(0,\sqrt{a(a+1)}\bigr],

$$and the inequality$$

holds for every $r\in(0,1)$ if and only if $a\in[3/7,1)$. The conjecture gives sharp parameter ranges for the two inequalities posed in the paper; the source notes that the first inequality fails for sufficiently small $r$ when $b_1b_2>a(a+1)$ and presents the assertion as suggested by computation, with no proof supplied here.