A 7-adic recurrence congruence for generalized Frobenius partitions with parameter 6

For integers $\alpha>0$, $\beta\in\{0,1,2,3\}$, and $n$ satisfying

$$\beta^2-6\equiv 12n\pmod{7^{2\alpha-1}},$$

The recurrence congruence conjecture.

$$c\psi_{6,\beta}(49n+24-4\beta^2)\equiv x_\alpha\,c\psi_{6,\beta}(n)\pmod{7^\alpha},$$

where $x_\alpha$ is a constant depending on $\alpha$. This predicts a family of 7-adic congruence relations among generalized Frobenius partition functions and is presented as another example of the congruence phenomenon discussed in the paper. Its resolution is not indicated in the supplied text.