I would give a slightly different answer for R4.
Each of the current mirror legs share the same basic base voltage, so except for different collector voltages and the Early effect, they should have the same collector currents. For all strings but the controlling string, the collector current is the LED current.
For the controlling string (leftmost), the base current needed for the follower between collector and base "steals" from the LED current, so there is a difference between the LED current and the collector current. The leftmost string should always have a slightly lower current than the LED currents in the rest of the legs (except for Early effect issues).
Of course, this error is helped greatly by having that follower there. It's reduced by the beta of that transistor.
Just like those small ballast resistors in the emitter legs of each of the mirrors helps to even out the effects of the LED I-V curves, this R4 helps to ballast the error between the controlling leg and the other legs. It adds a fixed amount of current which is somewhat larger than the sum of all the other base current contributions to the follower current, so the follower current's emitter current is not so sensitive to the actual base current, which can change with temperature, the number of current mirrors wired up, etc.). It does increase the magnitude of the error, but it lessens the variation in that error.
I don't agree with JS on danger w/o this R4 causing the bias current to be somehow redirected into the bases of the other legs causing it to "saturate them or go out of control". That doesn't make sense to me.
TL;DR R4 is there for reproducibility in results. It increases the error slightly between the left, controlling leg and the mirrored legs, but makes that error more constant over operating conditions.