  We introduce a class of subproduct systems of finite dimensional Hilbert
spaces whose fibers are defined by the Jones-Wenzl projections in
Temperley-Lieb algebras. The quantum symmetries of a subclass of these systems
are the free orthogonal quantum groups. For this subclass, we show that the
corresponding Toeplitz algebras are nuclear C$^*$-algebras that are
$KK$-equivalent to $\mathbb C$ and obtain a complete list of generators and
relations for them. We also show that their gauge-invariant subalgebras
coincide with the algebras of functions on the end compactifications of the
duals of the free orthogonal quantum groups. Along the way we prove a few
general results on equivariant subproduct systems, in particular, on the
behavior of the Toeplitz and Cuntz-Pimsner algebras under monoidal equivalence
of quantum symmetry groups.
