Additive approximation to bin packing

The bin packing problem is an NP-hard optimization problem that has many constant-factor approximation algorithms. I am looking for an additive approximation. I.e., given a set $I$ of items and bin capacity $B$, find a packing in which the number of bins is at most $OPT(I,B)+d$, where $OPT(I,B)$ is the optimal number of bins and $d$ is some small constant (ideally $d=1$).

Here I found an algorithm that finds a packing in at most $OPT(I,B)+O(log{OPT(I,B)})$ bins, but not $OPT(I,B)+$constant. This leads me to suspect that it may be impossible unless P=NP. But I did not find a proof of impossibility either.

Is such an additive approximation possible in polynomial time?