Metric TSP with integer edge cost

Given a metric TSP with integer edge cost upper-bounded by a constant $$C_{\max}$$, can we find an poly-time algorithm solving this TSP instance?

Indeed, even for TSP instances where all distances are $$1$$ or $$2$$ (note that these automatically satisfy the triangle inequality), Engebretsen and Karpinski showed that it is NP-hard to approximate TSP within a factor of $$\frac{741}{740} - \epsilon$$, for any $$\epsilon > 0$$.