Valuation growth for even overpartition tuples

Prove that for every even integer m≥2, with i_m=1 when 4 divides m and i_m=2 when m≡2 mod 4, and with R_i^(m) defined by the even-m recursion in Section 6, the quantities v_i=ν(R_i^(m)) satisfy, for every i>i_m, ν((R_i^(m))'(1))=v_i, ν(R_i^(m)(1))=v_i+1, and v_{i+1}=v_i+3.

Background

For even m, the paper studies the differences $\mathcal D_i^{(m)}=U^i(A^m)-U^{i-1}(A^m)$, where $A$ is the generating-function series for overpartition tuples with odd parts and $U$ extracts coefficients indexed by even integers. Unlike the odd-m case, the iteration is expressed entirely through polynomials in $\xi=A^4$. The authors define polynomials $R_i^{(m)}$ by $\mathcal D_i^{(m)}=(\xi-1)R_i^{(m)}(\xi)$ and establish the recursion $R_{i+1}^{(m)}=\bigl(R_i^{(m)}\bigr)$.

The relevant recursion satisfies $$\bigl(R_i^{(m)}\bigr)(1)=4R_i^{(m)}(1)+8(R_i^{(m)})'(1)$$. The two terms have the same apparent 2-adic scale, so the authors conjecture that cancellation produces an additional power of 2 at each iteration, yielding a three-power-of-2 increase in $v_i$. If true, this would imply internal congruences for even m with moduli growing like $2^{3i+c_m}$, with the modulus attained at n=1.