3D Lovelock gravity and the holographic $c$-theorem

We show that the scalar-tensor theory that arises in a rigorous $D\to 3$ limit of Lovelock gravity up to cubic order admits a holographic $c$-theorem and verify that the value of the $c$-function at the UV fixed point matches with the Weyl anomaly coefficient (up to an irrelevant factor). The constructed $c$-function is the same as one that follows from the "naive limit," which is obtained by scaling the couplings by a factor of $\frac{1}{D-3}$, and then setting $D=3$.