Let $p$ be a prime and consider the ($p$-deprived) Hecke algebra $\mathbb{T}$ which the projective limit of the Hecke $\mathbb{Z}_p$-algebras $\mathbb{T}_k$ which act on modular forms of level $1$ with coefficients in $\mathbb{Z}_p$ and of weight at most $k$. (See $\S$2.1 of https://www.math.uchicago.edu/~emerton/pdffiles/padic.pdf)

Is the Krull dimension of $\mathbb{T}$ known when $p = 2$ or $3$? Is anything conjectured? The only references I can find deal with cases where the corresponding residual representation is absolutely irreducible, which excludes these cases.