Congruences modulo powers of $3$ for generalized Frobenius partitions $C\Psi_{6,0}$

In 1984, Andrews introduced the family of partition functions $c\phi_k(n)$, which counts the number of generalized Frobenius partitions of $n$ with $k$ colors. In previous work, we proved a conjecture on congruences for $c\phi_6(n)$ modulo powers of 3. In this paper, we consider the $(6,0)$-colored Frobenius partition functions $c\psi_{6,0}(n)$. We establish a connection between the generating functions of $c\psi_{6,3}(n)$ and $c\psi_{6,0}(n)$ via an Atkin-Lehner involution, and prove congruences modulo powers of 3 for $c\psi_{6,0}(n)$.