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I'm not asking about whether the Hartree-Fock method will always converge, but, if it does, it seems like Wikipedia is saying that it will always converge to the global minimum energy of the TISE.

Minimising the expected energy is, in general, is not an easy task, since we are looking for a global minimum and finding the zeroes of the partial derivatives of $\varepsilon$ over all $\alpha+i$ is not sufficient. If $\psi(\alpha)$ is expressed as a linear combination of other functions ($\alpha_i$ being the coefficients), as in the Ritz method, there is only one minimum and the problem is straightforward. There are other, non-linear methods, however, such as the Hartree–Fock method, that are also not characterized by a multitude of minima and are therefore comfortable in calculations.

Is this true and if so, how?

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