Internal congruences modulo powers of $2$ for overpartition tuples with odd parts

Let $\overline{\mathrm{OPT}}_m(n)$ denote the number of overpartition $m$-tuples of $n$ into odd parts. We prove that for every odd $m\ge1$ and every $i\ge3$, \[\sum_{n\ge0}\Bigl(\overline{\mathrm{OPT}}_m\bigl(2^in\bigr)-\overline{\mathrm{OPT}}_m\bigl(2^{i-1}n\bigr)\Bigr)q^n \equiv 2^{\,i+1}\sum_{k\