Given $\mathsf{c\geq1}$, what is the probability that if you choose $\mathsf{A,B,\alpha\in\Bbb N}$ such that $\mathsf{A,B<\alpha<AB}$ holds we will have both $$\mathsf{gcd(A,B)=1}$$$$\mathsf{(AB)^{-1}\bmod\alpha<\log^c\alpha}\mbox{ or }\mathsf{(A^{-1}B)\bmod\alpha<\log^c\alpha}$$ satisfied?

Condition $\mathsf{gcd(A,B)=1}$ has probability $\frac{6}{\pi^2}$ asymptotically.

What is the probability if $\mathsf{A,B<\alpha<AB}$ is not enforced?