Conjecture on linear independence of trivial links in the fourth skein module

Let $T_i$ denote the trivial link with $i$ components, and let $\mathcal{S}_4(S^3)$ be the fourth skein module with parameters $a,b_0,b_1,b_2,b_3$, where the relevant inverses exist.

Linear-independence conjecture. The trivial links $T_i$ are linearly independent in $\mathcal{S}_4(S^3)$ in either of the following cases:

1. $a^4=1$, $b_3=-a^3b_0$, and $b_1=-a^3b_2$;
2. $ab_0^3=-b_3^3$, $b_2b_0^{-1}=b_3^2b_0^{-2}+b_3^{-2}b_0^2$, and $b_0b_1=b_2b_3$.

The conjecture would provide the linear independence needed for polynomial invariants arising from the fourth skein module; the supplied text gives special related results but leaves these parameter cases as conjectural.