Strictly singular operators in Tsirelson like spaces

For each $n\in \mathbb{N}$ a Banach space ${\mathfrak{X}}_{0,1}^n$ is constructed is having the property that every normalized weakly null sequence generates either a $c_0$ or ${\mathrm{\ell }}_1$ spreading models and every infinite dimensional subspace has weakly null sequences generating both $c_0$ and ${\mathrm{\ell }}_1$ spreading models. The space ${\mathfrak{X}}_{0,1}^n$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of ${\mathfrak{X}}_{0,1}^n$, for every $S_1,S_2,\dots ,S_{n+1}$ strictly singular operators on $Y$, the operator $S_1{S}_2\cdots S_{n+1}$ is compact. Moreover, for every subspace $Y$ as above, there exist $S_1,S_2,\dots ,S_n$ strictly singular operators on $Y$, such that the operator $S_1{S}_2\cdots S_n$ is non-compact.