The finite-freeness and specialization conjecture for the big $P$-ordinary Hecke algebra

Let $R$, $\kappa$, $\tau$ and $K_r$ be as above, and assume the independence-of-weight conjecture. Let $\Lambda_R^\circ$ be the relevant weight algebra, let $\Lambda_{R,\epsilon}^\circ$ denote its localization at the maximal ideal defined by a tame character $\epsilon$, and let $\mathbf{T}^{P\text{-ord}}_{K^p,[\kappa,\tau],R,\epsilon}$ be the corresponding localization of the big $P$-ordinary Hecke algebra.

Big $P$-ordinary Hecke algebra conjecture.

1. For each tame character $\epsilon$, $\mathbf{T}^{P\text{-ord}}_{K^p,[\kappa,\tau],R,\epsilon}$ is finite free over $\Lambda_{R,\epsilon}^\circ$.
2. If $\kappa$ is a $P$-very regular weight, $\kappa_p$ is its corresponding $p$-adic weight, and $I_\kappa$ is the kernel of the homomorphism $\Lambda_P^\circ\to R\subset\mathbb{C}_p$ induced by the restriction of $\kappa_p$ to $Z_P^\circ$, then the natural homomorphism

$$\mathbf{T}^{P\text{-ord}}_{K^p,[\kappa,\tau],R}\otimes\Lambda_R^\circ/I_\kappa\longrightarrow\mathbf{T}^{P\text{-ord}}_{K_r,\kappa,[\tau],R}$$

is an isomorphism.

This conjecture predicts both finite freeness over weight space and recovery of classical $P$-ordinary Hecke algebras by specialization at very regular arithmetic weights. It is conditional on the preceding independence-of-weight conjecture.