(I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.)

Consider the Temperley-Lieb algebra on $n$ strands, $TL_n$, viewed as an algebra over the base ring $\mathbb{Z}[\delta]$. The closed loop evaluates to $-\delta$. Thinking of $\delta$ as the quantum number $[2]$, one can view any quantum number $[m]$ as a polynomial in $\delta$. For any extension $\Bbbk$ of $\mathbb{Z}[\delta]$, we write $TL_{n,\Bbbk}$ for the corresponding algebra after base change.

The Jones-Wenzl projector $JW_n$ (if it exists) is the unique element of $TL_{n,\Bbbk}$ which is orthogonal to every cup and cap, and whose coefficient of the identity diagram (in the basis of crossingless matchings) is $1$. My first (and hardest) question is:

For which $\Bbbk$ does $JW_n$ exist?

This seems like a question for which the answer should be in the literature, but for the life of me I can not find it. I would like a precise answer! Is there an obvious representation-theoretic reason?

For example, the recursive formulae for the Jones-Wenzl projector imply that it will certainly exist when $[k]$ is invertible in $\Bbbk$, for all $k \le n$. Commonly in the literature, one is concerned with the case when $\Bbbk = \mathbb{C}$ and $\delta$ is specialized to $q+q^{-1}$ for a primitive $2m$-th root of unity $q$. In this case, $[k]$ is invertible for all $k < m$, and $[m]=0$. It is not hard to see that $JW_m$ does not exist.

However, in this case, the literature does not seem to state which Jones-Wenzl projectors exist for $k>m$! For example, when $[2]=0$, $JW_3$ is still well-defined (and some of its coefficients will vanish).

So my second question is:

When q is a primitive $2m$-th root of unity, which Jones-Wenzl projectors exist?

A more algebraic version of the question is:

Suppose that $[m]=0$. Note that it is entirely possible that $[k]=0$ (or is non-zero, but non-invertible) for $k<m$, when $k$ and $m$ are not relatively prime. Does $JW_{m-1}$ exist? If not, what additional conditions are required for it to exist?