Media converter on one end, cisco on the other - will it work or not?

Question

Computer networks

Now: there is optics, at both ends there are media converters (I don’t remember the exact model, something non-branded like links) with sfp modules.
It is planned: network upgrade, installation of cisco 2960 switches, installation of existing sfp modules in them.
Problem: There is no way to change hardware at both ends at the same time. If I put a tsiska on one end of the fiber and stick an sfp module from the converter into it, and the converter remains on the second end, will it work or not? On the one hand, it seems like it should, on the other hand, I asked the provider's engineers on occasion, and they talked about some problems - either because of buffering, or because of some other (in)compatibility.