New York must be really weird to reducer assessments to match a lien.
Sounds like theft to me.
In order for them to foreclose on arrears, the back taxes need to be a significant portion of the current assessed value. Once the amount of the arrears equals the assessed value, then they can foreclose on the property.
I use this method on a seasonal home in NY to keep the taxes and assessment stable.
The towns assess waterfront, and especially seasonal(s) at decidedly different rates because they know you don’t vote in that district.
They raise the assessment and thus my taxes, I refuse to pay for a couple of years and they lower them down to where they should be. I negotiate the arrears back down to the third assessment level, and pay them in full and the loop starts all over again. I have managed to keep my assessment flat to what I paid for the property 20 years ago.
The town assessor, town clerk, and the county clerk know me on a first name basis. They know, that I know what they are doing and I won’t stand for it. Tax Grievance Day is a waste of time and they will only give you a token reduction, but your assessment still went up. In most cases higher than what the real world market value is.