Do you continue shots when a patient transfers under your care and has been receiving less than effective doses of AIT, but insist they have been effective?

Yes, I'd continue the shots. For some patients, even lower doses produce a meaningful desensitization and build tolerance. At some point, I would recommend retesting his allergies to objectively confirm the efficacy of the shots.

The important thing to remember is that all answers to this question (including mine) are and will be largely based upon personal opinion which (hopefully) comes from clinical experience. There is no objective data that can shed enough meaningful light on this question to generate any sort of consen...

It depends. If the patient says IT has been effective, can we claim that they received "ineffective" doses (presumably meaning less-than-recommended doses)? And how long has the patient been on AIT? As noted by Dr. @Dr. First Last, it might be time to do a trial discontinuation, anyway.

I would not ...

My opinion is that too low of a dose increases the risk of relapse. I don't want that on my watch.

Notice some people doing shots don't share their extract, which makes me suspicious.