Non-singularity of the reduced Information Bottleneck operator

Let $BA_\beta$ be the Information Bottleneck operator in log-decoder coordinates $(\log p(y|\hat{x}),\log p(\hat{x}))$, and let $D_{\log p(y|\hat{x}),\log p(\hat{x})}BA_\beta$ denote its Jacobian at a solution of the IB beta ODE. A reduced IB root is an IB root represented using its effective cardinality.

Non-singularity conjecture. The Jacobian matrix

$$I-D_{\log p(y|\hat{x}),\log p(\hat{x})}BA_\beta$$

at the IB beta ODE is non-singular at reduced IB roots whenever it is well-defined, except perhaps at points of bifurcation.

This conjecture is motivated by the corresponding result for the rate-distortion Blahut–Arimoto operator after reduction to a root of finite support. Its status is not established in the supplied source; possible failure at bifurcation points is explicitly left open.