Stopping rules for cross-sectional phone surveys?

Hi, I am part of a group who is deploying a phone survey. We're stratifying by age and sex. The sampling frame doesn't exist so we're using RDD to select phone numbers and then shuttling folks into their age and sex strata when they respond.

We expect difficulty in reaching the stratum sample size for the oldest age and sex stratum.

What I was wondering was - are there accepted methods for stopping a survey when it's difficult to attain the desired sample size?

Is it as simple as "The survey ends in X days no matter what"?

Or, is it a matter of looking at estimates from the survey questions periodically and once they have "stabilized" then it's cool to stop? Of course, we would only look at the estimates in that age/sex stratum.

I saw some methods that essentially do that but everything I found is for longitudinal surveys and many seem to have some ability to attach census demographics to the phone numbers in order to do some imputation. We won't have that luxury and our survey is cross-sectional. But, maybe there's something similar for my kind of survey?