The stronger congruence-subgroup relation conjecture

Let $\Gamma^a$ and $\Gamma^b$ be the congruence subgroups in the preceding Hauptmodul conjecture, and suppose

$$\Gamma^a=\Gamma_{l_1}(r_1),\qquad \Gamma^b=\Gamma_{l_2}(r_2),$$

where $l_1,l_2\in\{0,1\}$. The notation $n_2$ denotes the exponent associated with the relevant Newton-polynomial parameter.

Stronger congruence-subgroup conjecture. Then

$$l_1=l_2,\qquad r_1=|n_2|r_2.$$

This is presented as a strengthening of the preceding conjecture and refines the relation between the two congruence subgroups. Its general validity remains unproved in the supplied text.