A family of modulo 128 congruences for overpartitions with 6-regular nonoverlined parts

Let $\overline{R_6^\ast}(n)$ denote the overpartition function in which the nonoverlined parts are $6$-regular. Modulo 128 congruence family. For all $n \geq 0$ and $k \geq 1$, we have

$$\overline{R_6^\ast}\left(18 \cdot 3^{2k+1} n + \dfrac{153 \cdot 3^{2k} - 1}{4}\right) \equiv 0 \pmod{128}.$$

This extends the observed modulo $8$ congruence to a family of congruences modulo $128$, building on the preceding conjectured case.