Strict complete monotonicity conjecture for $F_3$

Let $F_3$ be the function defined in Theorem 3 of the source paper. A function is strictly completely monotonic on an interval if all of its derivatives have alternating signs there, namely $(-1)^kF_3^{(k)}(x)>0$ for every integer $k\geq 0$ and every $x$ in the interval. Strict complete monotonicity conjecture. The function $F_3$ is strictly completely monotonic on $(0,1/2]$. If true, this would yield sharp lower and upper bounds for $R(x)$ expressed in terms of $B(x)/[1+x(1-x)]$ and polynomials.