The uniqueness conjecture for tetrahedrally symmetric monopoles

A symmetric monopole is described by the function $Q_0(z)$, whose denominator is an elliptic-function expression associated with the monopole's spectral curve; $n_1$ denotes the integer parameter indexing the symmetric monopole.

Uniqueness conjecture. For a symmetric monopole, the denominator of $Q_0(z)$ has $2(|n_1|-1)$ zeros, and consequently the tetrahedrally symmetric monopole is the only monopole in this class.

This conjecture is suggested by extensive numerical calculations following the analysis of symmetric monopole curves. It asserts that the additional zeros generally obstruct the required monopole conditions, leaving only the tetrahedrally symmetric case.