Equality of quantum and classical learning coefficients on affine open subsets

Under the fundamental conditions, let $K$ and $K^Q$ denote the classical and quantum average log loss functions, respectively. Let $k$ and $k^Q$ be the corresponding learning coefficients computed from their resolutions on an affine open subset of the parameter space.

Quantum–classical learning-coefficient conjecture. The functions $K$ and $K^Q$ behave similarly under the fundamental conditions; equivalently,

$$k=k^Q$$

on any affine open subset.

This conjecture connects quantum information theory with algebraic geometry by asserting equality of the classical and quantum learning coefficients without the global holomorphic extension used in the preceding proposition. The supplied source gives no resolution of the conjecture.