Why increasing the beam width does not necessarily improve the beam search solution?
Imagine your tree looks as follows:



You are looking for the leaf with minimum label and the label of each internal nodes represents the heuristic value associated with the subtree routed at that node.

If the width of your beam is just $1$, you start at the root and then go $\rightarrow 5\rightarrow7\rightarrow0$.

If the width of your beam is $2$, you first take $[5,6]$ as potential candidates. Look at $5's$ children, so your best candidates at that point for the next level are $[7,8]$. Look at $6's$ children, both better candidates, so your potential candidates become $[3,4]$. Then you'll find only $9$'s going down, which is worse than the solution found with a beam width of $1$.

But if your heuristic is well behaved enough, you might be able to guarantee that increasing the beam width can not degrade the quality of the solution that is found.