Conjecture: $\pi\le\sum_{cyc}\operatorname{arcsec}(f(a,b,c))\le3\pi/2$, where $f(a,b,c)=\frac{a^2+b^2-c^2-a^2b^2c^2}{2ab(1-c^2)}$ and $a,b,c>\sqrt{3}$

Question

I have the following suspicion connected with non-euclidean geometry:

Let $a,b,c>\sqrt{3}$, and $$ f\left(a,b,c\right)=\frac{a^{2}+b^{2}-c^{2}-\left(abc\right)^{2}}{2ab\left(1-c^{2}\right)}. $$ Is it true, that $$ \pi\leq\text{arcsec}\left(f\left(a,b,c\right)\right)+\text{arcsec}\left(f\left(b,c,a\right)\right)+\text{arcsec}\left(f\left(c,a,b\right)\right)\leq\frac{3\pi}{2}, $$ where $\text{arcsec}\left(x\right)$ is the inverse function of $\text{sec}\left(x\right)$?