Quantum regularity inherited from an associated classical model

Let $(\rho,\sigma(\theta))$ be a quantum model and let $(q(x),p(x\mid\theta))$ be its associated classical model. A quantum model is quantum regular when its quantum parameter set has a unique minimizer, the Hessian of the quantum relative entropy at that minimizer is positive definite, and the minimizer has an open neighborhood in the parameter space. The associated classical model is called regular when it satisfies the corresponding classical regularity conditions.

Quantum regularity conjecture. If $(q(x),p(x\mid\theta))$ is regular, then $(\rho,\sigma(\theta))$ is also quantum regular.

This conjecture proposes that introducing measurements and passing to the associated classical model cannot make a regular quantum model singular. Its resolution is not established in the supplied source.