The LM317L can be used as a reference.
There are also transistor array, such as the MMPQ3906 and MMPQ2907A, which have better matching than discrete parts.
I think I see how you expect it to work, and it seems to have been cleverly engineered!
Also neat idea to use cheap, transistor arrays, to get nicely matched transistors.
But the LM317L's, can be a bit tricky. So I raise the following possible concerns (friendly technical comments hopeful, NOT argument starting).
Minor thing, I think R3 .. R6, would have around 600mV, so around 27mA, but the LEDs want (a max of) 20mA. So, perhaps 30 Ohms or more, might be better.
The LM317L
https://www.ti.com/lit/ds/symlink/lm317l.pdf seems to need at least 2.5 volts across it, input to output terminals.
VI – VO Input-to-output voltage differential, which seems to be specified as 2.5V minimum.
But it also needs around 1.2V to be across R7 470 Ohms, to create the minimum 2.5mA output current (for it to be regulated).
Which would be rather tight, because the 5V rail may be a bit under, Q1 (assuming saturated), may still have a bit of voltage drop, and then Q6, could be more like 0.7V (transistors tend to be doped (complicated subject area, but needed to make it a good transistor) to have ever so slightly higher Vbe voltages, than typical small signal diodes).
2.5V (VI-VO) + 1.2V (to get current via R7) + 0.1V Q1 (could be less or more) + Q6 Vbe 0.7 + R1 220 Ohms (at desired current), 0.6V = 5.1V
N.B. Opinions on some of the figures I just estimated, can easily vary. E.g. Some will say Vbe is 600mV, also some might say Vsat on Q1 is zero, etc.
But anyway, my estimate seems to be it would need 5.1V or more on the 5V rail, and good design should allow a 5% or 10% drop on the 5V rail, and it should still be expected to work well.
Sorry if I seem to be nit picking.
Also the LM317L can be a bit tricky to analyse (for me at least), so I readily accept I could be mistaken.
EDIT: Where I have doubts in my criticism. Is can the adjustment pin of the LM317L, count as the minimum current source. If it can, then you could be good to go, or does it have to come from the output pin, which the wording of the datasheet seems to imply (to me).
EDIT2: Because the currents are so low, and the datasheet bit I mentioned, is a recommendation, rather than an essential requirement. My estimates could easily be too high, which could easily make the real minimum voltage, perhaps 4.8V (wild estimate), which would begin to be fine.