Baricz's conjecture on parameter monotonicity of Gaussian hypergeometric functions

Let $F(a,b;c;r)={}_2F_1(a,b;c;r)$ denote the Gaussian hypergeometric function, defined for the parameters under consideration by

$$F(a,b;c;r)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n n!}r^n.$$

For $a>0$, $b>0$, and $r\in(0,1)$, consider the ratio of contiguous Gaussian hypergeometric functions. Baricz's conjecture. For each $a>0$ and $r\in(0,1)$, the function

$$b\longmapsto \frac{F(a+1,b+1;a+b+1;r)}{F(a,b;a+b;r)}$$

has negative derivative. In particular, for all $a,r\in(0,1)$,

$$\frac{F(a+1,2-a;2;r)}{F(a,1-a;1;r)}>\frac{F(a+1,5/2-a;5/2;r)}{F(a,3/2-a;3/2;r)}.$$

The conjecture concerns Turán-type inequalities and parameter dependence for zero-balanced Gaussian hypergeometric functions, with generalized elliptic integrals as important special cases. The source attributes it to Baricz; no resolution is supplied here.