A path metric is a graph formed by a path $P$ of $n.$ Edges that form $P$ have potentially

different weights. Meanwhile, any other edge has a weight that equals to the weight of the

shortest path between its endpoints on $P.$ For example, in the figure below, the weight of

the edge between $b$ and $f$ is $2+2+4+1=9.$

A clever student named Eva claims that the MST algorithm always produces the optimal

TSP tour when the input graph is a path metric. Prove or disprove Eva's claim.