Interior angle sum conjecture for horizontal-like geodesic triangles in Sol geometry

Let a horizontal-like geodesic triangle in Sol geometry have interior angles $\omega_1(a),\omega_2(a),\omega_3(a)$, with parameter $a\in\mathbf{R}^+$. Define its interior angle sum by

$$\Omega(a)=\sum_{i=1}^3\omega_i(a).$$

Interior angle sum conjecture. The sum of the interior angles of any horizontal-like geodesic triangle is greater than $\pi$:

$$\Omega(a)>\pi.$$

The surrounding discussion gives this result for the horizontal-like isosceles family and reports that $\Omega(a)$ is strictly increasing, with limiting values $\pi$ as $a\to0$ and $3\pi/2$ as $a\to\infty$. The parser supplies no evidence that the stated claim has been resolved beyond this family.