If the conformal equivalence class already encodes metric...

Conformal equivalence class(as the set of related geometric forms): A group of different geometric shapes that all have the same angles as each other, even if their sizes differ; they're all equivalent in terms of angle-relationships.
Geometric underdetermination(as used in philosophy of geometry and physics): A situation where the available information about a shape or space doesn't uniquely pin down all its properties—there are multiple ways to describe it that fit the same data.
Metric ratios(as used in geometry and physics): The numerical relationships between distances or sizes in a space—essentially, the proportions that tell you how big or far apart things are relative to each other.
Scalar factor(as used in mathematics and physics): A single number you multiply by to change the size of something uniformly, without changing its shape—like zooming in on a photograph.
Structural absence(as used in metaphysics and philosophy of science): Something that genuinely doesn't exist or isn't really there, as opposed to just being unknown or hard to measure.
length connection(Term adopted in the passage to replace Weyl's own usage of 'metric connection' in order to avoid confusion with modern terminology): An additional geometric structure, expressed as a gauge field A_j, that governs the congruent displacement of lengths and, together with the conformal structure, determines a unique symmetric linear connection.

2 topics

If the conformal equivalence class already encodes metric ratios up to a scalar factor, the demand for an independent length connection conflates geometric underdetermination with genuine structural absence.