What experimental evidence supports the Schrödinger equation as a fundamental postulate?

I understand that the Schrödinger equation is usually introduced as a postulate of non-relativistic quantum mechanics. However, I often see it motivated or "derived" from classical ideas such as energy, momentum, and the de Broglie relations. My question is: what is the actual experimental justification for taking the Schrödinger equation as a fundamental law? Is there an experiment that directly distinguishes the Schrödinger equation from other possible equations of motion, or is its justification mainly that it successfully reproduces experimental results across many different systems? I am asking specifically about the physical/experimental justification, rather than a mathematical derivation.