Joint distribution in residue classes of the base-$q$ and Ostrowski digital sums

Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For $$\alpha=[0;\overline{1,m}],\ m\geq 2,$$ let $S_{\alpha}(n)$ denote the sum of digits in the Ostrowski $\alpha$-representation of $n$. Let $m_1,m_2\geq 2$ be integers with $$\gcd(q-1,m_1)=\gcd(m,m_2)=1.$$ We prove that there exists $\delta>0$ such that for all integers $a_1,a_2$, \begin{eqnarray*} &&|\{0\leq n<N: S_{q}(n)\equiv a_1\pmod{m_1},\ S_{\alpha}(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-\delta}). \end{eqnarray*} The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case $\alpha=[\ \overline{1}\ ]$ by Spiegelhofer.