Conjectural modulo-7 congruences for overcolored partition k-tuples

Prove the conjectured modulo-7 congruences \(\bar{b}^4_{2,3}(14n+7)\), \(\bar{b}^2_{5,4}(14n+7)\), and \(\bar{b}^3_{5,4}(14n+7)\) vanish modulo 7 for all \(n\ge0\).

Background

The paper lists three computationally motivated congruences modulo 7 for overcolored partition k-tuples. They are stated in a conjecture environment and are not established in the paper.

References

\begin{conjecture} For all $n\ge 0$, \bar{b}4_{2,3}(14n+7)\equiv 0\pmod{7},\quad \bar{b}2_{5,4}(14n+7)\equiv 0\pmod{7},\quad \bar{b}3_{5,4}(14n+7)\equiv 0\pmod{7}. \end{conjecture}

— Overcolored Partition $k$-tuples Restricted by Parity of the Parts  (2609.03926 - Thejitha et al., 3 Sep 2026) in Section 6s7, Concluding remarks, item 3, second conjecture