Is a $CD(K,\infty)$ space a length space?

Is a $CD(K,\infty)$ space a length space?

Let $(X,d)$ be a complete and separable metric space endowed with a
nonnegative Borel measure $\mu$ with support $X$ and satisfying
\begin{eqnarray} \mu(B(x,r))<\infty,\quad\mbox{for every }x\in X\mbox{ and
}r>0, \end{eqnarray} where $B(x,r)$ is the ball of radius $r$ centered at
$x$ in $X$ w.r.t. the metric $d$.
Let $(X,d,\mu)$ be a $CD(K,\infty)$ space in the sense of Sturm. Is
$(X,d)$ a length space?
Many thanks.