Ingham divisor problem with square-free numbers

For the number $N(x)$ of solutions to the equation $aq-bc=1$ in positive integers $a,b,c$ and square-free numbers $q$ satisfying the condition $aq\leqslant x$ the asymptotic formula
$$ N(x)=\sum_{n\leqslant x}2^{\omega(n)}\tau(n-1)=\xi_0 x\ln^2 x + \xi_1 x\ln x + \xi_2 x + O(x^{5/6+\varepsilon}) $$
is obtained for any $\varepsilon>0$, where $\xi_0,\xi_1,\xi_2$ are constants.