YOLOv3 Predicting bounding boxes for grid containing multiple centers
It depends on the combination of thresholds of the objectness score and non-max suppresion (NMS):
Objectness score $C$
Apart from the location and dimensions predictions ($t_x, t_y, t_h, t_w$) for each box, an objectness score $C$ is also predicted for each of the boxes. It tells us how likely an object is inside  of a certain predicted bounding box (BB).
But How much it needs to be to ensure an object is inside?
This has not a precise answer, instead we use a certain threshold. For example, we could set that a BB contains an object if and only if the objectness $C&gt;0.5$ (threshold = $0.5$). This value depends on the implmentation and it can be asked to the user as in this repo
Non-max suppresion (NMS)
Now that we know that there is an objectness score threshold, we can rephrase the question  in the following way:
What if two BBs of the same grid cell have an objectness score $&gt;$ threshold? Would one of these two BBs be eliminated?
This depends on the threshold used in non-max suppression. If this concept is new to you, I recommend you this tutorial about it.
Broadly speaking it's used to discard bounding boxes that have a lower objectness score than the bounding box assigned to the same class that has the highest objectness score, and that have an overlap with it bigger than the threshold we are talking about now.
This means that higher thresholds used in NMS will lead to discarding a fewer number of boxes (we will be allowing a bigger overlapping between the boxes).
Conclusion
So, to sum up, In the situation of the question (more than one object per grid cell). Only boxes that have values of $C$ higher than its threshold will be considered. And out of these boxes, if there are some that are assigned to the same class then the box that have the highest $C$ will be considered and also the ones that don't get eliminated after applying non-max suppresion.