The conjecture on monotonicity of the zeta-function expression $F(s)$

Let $\zeta(s)$ denote the Riemann zeta function, and define

$$F(s)=2(s+1)(2^{s+2}-1)\zeta(s+2)-2\pi^2(2^s-1)\zeta(s),\qquad 1<s<\infty.$$

Monotonicity conjecture. The function $F(s)$ is strictly increasing on $(1,\infty)$. In particular, $s=2$ is the unique zero of $F(s)$.

The conjecture is motivated by the dependence on $q$ of $h_\infty”(1/2)$ and would remove the extra quantities $N(q)$ from the preceding convexity proposition if established.