I've pored over this for many hours now, and I can't disagree with anything anybody here has said, because the arguments are all solid. So, the biggest question remains, does OP's (@MRB's) circuit eliminate thermal runaway in one of the LEDs?
I had to simulate it, with CircuitLab, which I did for two cases. The original, unbalanced mirror, and a balanced version inspired by the superb suggestions from @Neil_UK.
I use multiple LEDs in series, to simulate a real life chain, and I model the combined drop in forward voltage of them all (due to LED temperature rise) using a voltage source, connected in such a way that as the source's voltage increases it reduces the combined potential difference across the whole chain. Note how the positive end of the source is at the bottom, something I had to do because CircuitLab inconveniently refuses to perfom DC sweeps in the positive-to-negative direction.
simulate this circuit – Schematic created using CircuitLab
Below are the plots of current and total power dissipation in the left and right chains (left is blue), where I calcluate power as the product of the total potential difference across each chain (including V1, the forward-voltage adjustment) and the current through it.
Each plot is a sweep of V1 from 0 upwards, resulting in a progressive reduction of total voltage drop, thereby representing a somewhat naive analogue of increasing temperature.
My interpretation of this is that up until about 400mV reduction in combined forward voltage of the left chain, the power in both left and right chains diminishes with temperature. That's enough to prevent thermal runaway in either chain, but it highlights a problem.
If either chain's combined forward voltage is greater than 400mV different from the other's, then this system enters a different regime, where further disparity will result in thermal runaway. Therefore, this circuit is not very effective if the LEDs are not well matched to begin with.
Under 400mV of temperature-induced forward voltage reduction seems to vindicate me and @jp314, who claimed that the circuit will work. However, in the far more likely scenario that the LEDs will not be well matched, @tobalt's opinion seems easily the better judgement.
Now I include resistors to mitigate the "master-slave" aspect of the mirror, as @Neil_UK suggests (though I do not include emitter resistors to combat transistor mismatch). Here's the schematic, with current and power plots:
simulate this circuit
This seems to solve the issue of LED mismatch, because as combined forward voltage continues to decrease in the left chain, well beyond the previous turning point, power in both chains continues to diminish.
In the light of this, it seems to me that the "balanced" current mirror will actually function to prevent thermal runaway in either chain.
I am aware that the way I modelled the LEDs' negative forward voltage temperature coefficient is pretty naive, and it won't surprise me if my approach is heavily criticised, but that's all I have. This thing already had my head spinning anyway, so I welcome anything that will put this to bed, even (and especially) if it completely invalidates all this.