The existence of a strict deformation quantization of X_k=S(M_k({mathbb {C}})), the state space of the ktimes k matrices M_k({mathbb {C}}) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. https://doi.org/10.1142/S0129055X20500312). In fact, since increasing tensor powers of the ktimes k matrices M_k({mathbb {C}}) are known to give rise to a continuous bundle of C^*-algebras over I={0}cup 1/mathbb {N}subset [0,1] with fibers A_{1/N}=M_k({mathbb {C}})^{otimes N} and A_0=C(X_k), we were able to define a strict deformation quantization of X_k à la Rieffel, specified by quantization maps Q_{1/N}:/ tilde{A}_0rightarrow A_{1/N}, with tilde{A}_0 a dense Poisson subalgebra of A_0. A similar result is known for the symplectic manifold S^2subset mathbb {R}^3, for which in this case the fibers A'_{1/N}=M_{N+1}(mathbb {C})cong B(text {Sym}^N(mathbb {C}^2)) and A_0'=C(S^2) form a continuous bundle of C^*-algebras over the same base space I, and where quantization is specified by (a priori different) quantization maps Q_{1/N}': tilde{A}_0' rightarrow A_{1/N}'. In this paper, we focus on the particular case X_2cong B^3 (i.e., the unit three-ball in mathbb {R}^3) and show that for any function fin tilde{A}_0 one has lim _{Nrightarrow infty }||(Q_{1/N}(f))|_{text {Sym}^N(mathbb {C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0, where text {Sym}^N(mathbb {C}^2) denotes the symmetric subspace of (mathbb {C}^2)^{N otimes }. Finally, we give an application regarding the (quantum) Curie–Weiss model.