Repdigit Niven-number parametrization conjecture

Let $a_{(k)}$ denote the $k$-digit repdigit formed by repeating the digit $a$, and let $a\in\{1,2,3,4,5,6,7,8,9\}$. A repdigit is a Niven number when

$$10^k\equiv 1\pmod{9ka}.$$

Define

$$m_0=37,\quad m_1=333667,\quad m_{21}=757,\quad m_{22}=440334654777631,$$ $$m_{31}=163,\quad m_{32}=9937,\quad m_{33}=2462401,\quad m_{34}=676421558270641,$$ $$m_{35}=130654897808007778425046117.$$

Repdigit Niven-number parametrization conjecture. The solutions of the repdigit Niven-number equation are parameterized by

$$k=k_{\{n\alpha\beta\gamma\delta\}}=3^n\cdot m_0^\alpha\cdot m_1^\beta\cdot m_{21}^{\gamma_1}\cdot m_{22}^{\gamma_2}\cdot m_{31}^{\delta_1}\cdot m_{32}^{\delta_2}\cdot m_{33}^{\delta_3}\cdot m_{34}^{\delta_4}\cdot m_{35}^{\delta_5},$$

where

$$\{\alpha,\beta,\gamma_1,\gamma_2,\delta_1,\delta_2,\ldots,\delta_5\}\in\mathbb{N}_0$$

and the minimal-index constraints are

$$n\geq 0\quad\text{if all exponents are zero},$$ $$n\geq 1\quad\text{if }\alpha\neq 0,\qquad n\geq 2\quad\text{if }\beta\neq 0,$$ $$n\geq 3\quad\text{if }\{\gamma_1,\gamma_2\}\neq\{0\},\qquad n\geq 4\quad\text{if }\{\delta_1,\ldots,\delta_5\}\neq\{0\}.$$

The parametrization is intended to demonstrate infinitely many repdigit Niven numbers, which are consequently permutation-invariant Niven numbers; the source does not provide evidence resolving whether this proposed parametrization is complete.