Finiteness of the image of the Reidemeister torsion of a splice

The set $RT(M)$ of values of the $SL(2,\mathbb{C})$-Reidemeister torsion of a 3-manifold $M$ can be both finite and infinite. We prove that $RT(M)$ is a finite set if $M$ is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and $A$-polynomials of knots.