Congruence families modulo powers of $7$ for $4$-colored generalized Frobenius partitions

In 2012, Peter Paule and Cristian-Silviu Radu proved an infinite family of Ramanujan type congruences for $2$-colored Frobenius partitions $c\phi_2$ introduced by George Andrews. Recently, Frank Garvan, James Sellers and Nicolas Smoot showed that this family of congruences is equivalent to the family of congruences for $(2,0)$-colored Frobenius partitions $c\psi_{2,0}$ introduced by Brian Drake and by Yuze Jiang, Larry Rolen and Michael Woodbury for the general case. Motivated by Garvan, Sellers and Smoot's work, Rong Chen and Xiao-Jie Zhu found modular transformations relating the $c\psi_{k,\beta}$ for fixed $k$ and varying $\beta$. As an example, they proved a family of congruences for $c\psi_{3,1/2}$ following Paule and Radu's work and then proved the equivalence between $c\psi_{3,1/2}$ and $c\phi_3=c\psi_{3,3/2}$. In the present paper, we give a new example of Chen and Zhu's framework for $c\psi_{4,\beta}$. Our proof is considerably simpler.