Example: traceless C*-algebra universally generated by projections

Question

Does someone have an example of

1. a non-zero C*-algebra which is
2. universally generated by
3. finitely many projections (not all commuting) together with a unit and plus
4. necessarily satisfying some additional relations such that
5. there remain no traces?

In other words, is there a non-zero quotient $$C^*(Z/2*\ldots*Z/2)\to B\to 0$$ which carries no traces?
So basically the question is about the representation theory of $C^*(Z/2*\ldots*Z/2)$.

Or more generally, what classes of groups are there with some traceless quotients?

EDIT: Question got answered on mathoverflow:
https://mathoverflow.net/a/409043/45494