Extension conjecture for theories with well-behaved thermodynamics

A theory with well-behaved thermodynamics is a general probabilistic theory whose thermodynamic structure satisfies the conditions assumed in the paper. A sharp theory with purification is a theory satisfying Sharpness and Purification, so that physical transformations and mixed states admit the corresponding pure, reversible dilations described in the framework.

Extension conjecture. Every theory with a “well-behaved” thermodynamics can be extended to a sharp theory with purification.

If true, this would provide formal foundations for classical thermodynamics by embedding classical systems into a broader theory where purification and entanglement are available. The conjecture is presented as a possible explanation and remains open in the supplied source.