Four-dimensional operator systems without the lifting property

The purpose of this note is to provide a family of explicit examples of $4$-dimensional operator systems contained in the Calkin algebra $\mathcal{Q}(\mathcal{H})$ on a separable infinite-dimensional Hilbert space $\mathcal{H}$ for which the identity map has no unital completely positive (ucp) lift to $\mathcal{B}(\mathcal{H})$ with respect to the canonical quotient map $\pi :\mathcal{B}(\mathcal{H})\to \mathcal{Q}(\mathcal{H})$. More specifically, to each unital $C^{\ast }$-algebra $\mathcal{A}$ generated by $n$ unitaries and unital $\ast$-homomorphism $\rho :\mathcal{A}\to \mathcal{Q}(\mathcal{H})$ with no ucp lift, we construct a four-dimensional operator subsystem $\mathcal{S}$ of $M_{n+1}(\mathcal{A})$ without the lifting property. As a result, for each $n\ge 2$ we exhibit a four-dimensional operator system $\mathcal{S}$ in $M_{n+1}(C_r^{\ast }({\mathbb{F}}_n))$ without the lifting property. We also obtain explicit examples where the generalized Smith-Ward problem for liftings of joint matrix ranges for three self-adjoint operators has a negative answer.