The independence-of-weight conjecture for the $P$-ordinary Hecke algebra

Let $R$ be the coefficient domain, let $\tau$ be a $P$-nebentypus, let $K_r$ be the level at $p$, and let $[\kappa]$ denote the $P$-parallel lattice associated with a dominant character $\kappa$.

Independence-of-weight conjecture. If $\kappa_1$ and $\kappa_2$ are dominant characters such that $[\kappa_1]=[\kappa_2]$, then there is a canonical isomorphism

$$\varprojlim_r \mathbf{T}^{P\text{-ord}}_{K_r,\kappa_1,[\tau],R}\xrightarrow{\sim}\varprojlim_r \mathbf{T}^{P\text{-ord}}_{K_r,\kappa_2,[\tau],R}.$$

This predicts that the inverse-limit $P$-ordinary Hecke algebra depends on the weight only through its $P$-parallel lattice. It is subsequently assumed in the paper when formulating the structure theorem for the big Hecke algebra.