Impact on garden birds when hawks are used to deter seagulls

A Harris hawk is being used to deter seagulls from nesting on a nearby block of flats.  I'm concerned that this scares the garden birds away too.  In response, the falconry company says:

"In fact, the opposite is true. Large hawks flown on the Estate will increase small songbird activity as Harris Hawks regularly flown on any Estate, will lower the number of crows, magpies and gulls which prey on songbirds and their nests. Large hawks are at the top of the food chain but do not prey on small birds. Cats pose the greatest risk to songbirds and small mammals."

I'm dubious but would love to hear an expert opinion on this.