Dishwasher Air Gap hookup size mismatch?

[stinkyDavis]

Hi,

I have a dishwasher but no garbage disposal. I want to install an air gap. It's my understanding that the standard sizes are:

• dishwasher discharge hose: 5/8" ID
• dishwasher wye sink tailpiece: 5/8" OD (takes 5/8" hose)
• air gap inlet: 5/8" OD (takes 5/8" hose)
• air gap discharge: ?

I just bought new dishwasher discharge hose, 5/8", and a dishwasher tailpiece, copper, 5/8" fitting. They had a number of these in plastic as well, all 5/8", and one straggler with a 7/8" fitting. I'm looking for an air gap with a 5/8" discharge fitting so that I can make the hookup directly from airgap to tailpiece with a single length of hose, using no adapters.

The only air gaps I can find have a 7/8" discharge fitting, which it seems is used with a garbage disposal. Some websites refer to this as "standard", so I'm thinking there must be a non-standard air gap for connection to a dishwasher tailpiece (5/8") when you don't have a disposal. But I can't find this kind of air gap.

Does anyone know of an air gap that meets this requirement? Is the only way to do this with a reducer? Am I nuts?