I thought peperomia too but usually the inflorescenses of peperomia are singular rather than one splitting into multiple. But that is why I asked if there were ever actual flowers on those spikes. If the answer is no, then they do seem to be inflorescences which would be indicative of peperomia of some sort.

Peperomia 'Rana Verde'

It looks a lot like doublefile (Shasta) from the branching, but I've never seen it bloom pink! Oh, I found this: 'Pink Beauty'

Interesting about the suckering. I thought those were just weeds. I don't recollect seeing suckers on Fagus sylvatica over here. It’s certainly not known for it. Does it sucker in the US? Does Fagus grandifolia sucker? Seeing leaves would still be helpful to pin down the species.