U. Simon posed the following conjecture in 1980: if f is an isometric minimal immersion from a Riemann surface M to an m-dimensional round sphere whose curvature is pinched between two consecutive terms of a given sequence, then its curvature is constant and M has to be a sphere. For now, this has been solved for very few cases and there are no counterexamples. I will first explain the background to understand this conjecture and give an overview of the solved cases. Then I will introduce a method, simple but still difficult to use in general, that gives intriguing results and can be used to prove the conjecture for many of the known cases.