Uniqueness conjecture for tetrahedral genus-zero dessins

Let a genus-$0$ passport be of one of the forms

$$\pi=(2^r,a \mathop{|} 3^s,b \mathop{|} 3^t,c),$$

or

$$\pi=(2^r,a \mathop{|} 3^s \mathop{|} 3^t,b,c),$$

where $2\nmid a$, $3\nmid b$, and $3\nmid c$. A realization of $\pi$ is a dessin d'enfant having this passport.

Uniqueness conjecture. For each such passport, its realization as a dessin d'enfant is unique.

The preceding theorem establishes existence of a realization for every passport in these two families. The uniqueness claim is presented as supported by computer calculations; no proof or resolution is given here.