Is the group $\mathrm{SL}(6,\mathbb{Z})$ $(2,3)$-generated?

The problem whether the group $\mathrm{SL}(6,\mathbb{Z})$ can be generated by an involution and an element of order three is considered. The problem is reduced to the question whether $\mathrm{SL}(6,\mathbb{Z})$ is generated by one of eight explicitly written pairs of matrices.