Therrin there are always significant ongoing and increasing maintenance issues with subordinated trees even small to medium sized ones. Now this doesn't necessarily mean they have to be removed...so long as the tree woner is made aware of what the most likely outcome down the track will be sure the tree can be managed.
You would expect much more deadwood production in a suppressed tree, somewhat weakened growth with greater suceptability to many pest and diseases due to reduced vigour.
The situation in your example is made more problematic by the proximity of the house..more deadwood over the roof..noise and debris, leaning growth to get away from the dominant tree...even mre leaf debris over the roof (not an issue at all for me but might be for the tree owner)
supression.JPG