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

Let ${\overline{\mathrm{O}\mathrm{P}\mathrm{T}}}_m(n)$ denote the number of overpartition $m$-tuples of $n$ into odd parts. We prove that for every odd $m\ge 1$ and every $i\ge 3$, $$\sum _{n\ge 0}\left({\overline{\mathrm{O}\mathrm{P}\mathrm{T}}}_m\right(2^{i}n)-{\overline{\mathrm{O}\mathrm{P}\mathrm{T}}}_m(2^{i-1}n\left)\right)q^n\equiv 2^{i+1}\sum _{k\ge 0}{q}^{(2k+1{)}^2}(\mathrm{m}\mathrm{o}\mathrm{d}{2}^{i+2})\mathrm{.}$$ Thus ${\overline{\mathrm{O}\mathrm{P}\mathrm{T}}}_m(2^{i}n)\equiv {\overline{\mathrm{O}\mathrm{P}\mathrm{T}}}_m(2^{i-1}n)(\mathrm{m}\mathrm{o}\mathrm{d}{2}^{i+1})$, with equality of $2$-adic valuations exactly at the odd squares. The proof is elementary and uniform in $m$: a single family of integer polynomials, given by a three-term recurrence, governs every $U$-operator identity involved, and a divisibility statement supplies one power of $2$ per iteration.