Conjecture on the trivial central-character skein module at roots of unity

Let $M$ be an oriented closed $3$-manifold, and let $\zeta$ be a primitive $2N$-root of unity with $N$ odd. Let $S_\zeta(M)_{[\mathbbm{1}]}$ denote the skein module component associated with the trivial representation, and let $X(M)$ be the relevant character space. For a knot $K$, write $E_K(r)$ for the Dehn filling of its exterior with slope $r$.

Trivial-character skein-module conjecture. If $X(M)$ is finite, then

$$S_\zeta(M)_{[\mathbbm{1}]}\simeq\mathbb{C}.$$

Consequently, for $K$ verifying the source's condition and for almost all $r\in\mathbb{Q}\cup\{\infty\}$,

$$S_\zeta(E_K(r))\simeq\mathbb{C}^{n},$$

where $n$ is the number of characters of $X(M)$ counted with multiplicity.

The conjecture concerns the remaining trivial-central-character summand after the decomposition of the skein module. The supplied text states that this summand is not yet known, while the claimed consequence applies to almost all Dehn fillings and roots of unity under the stated hypotheses.