Action of $\mathrm{M}(0,2n)$ on some kernel spaces coming from $\mathrm{SU}(2)$-TQFT

For $n$ an even number, we study representations of the mapping class group of the $n$-punctured sphere arising from $\mathrm{SU}(2)$-TQFT when all punctures are colored by the same integer $N \geq 1$. We prove that the conjecture of Andersen, Masbaum and Ueno holds for the $4$-punctured sphere for all $N \geq 2$. In the case $n \geq 6$ of punctures, we prove it for the pseudo-Anosovs satisfying a homological condition, namely they should act with a non trivial stretching factor on certain eigenspaces of homology of a $\frac{n}{2}$-fold branched cover considered by McMullen. The main idea is to consider the kernel space which is the kernel of the natural map from the skein module to the $\mathrm{SU}(2)$-TQFT. Our main theorem identifies, as representations of mapping class groups, certain of these kernel spaces with homology eigenspaces considered by McMullen. Our results concerning the AMU conjecture are obtained by taking appropriate limits of the quantum representations in which such a kernel space may appear as a subspace of the limit representation.