For a natural number $k$ (0 is a natural number), let $T_k$ be the collection of all languages that can be efficiently decided by quantum circuits consisting of Clifford gates and at most $k$ $T$-gates (that is, we modify the definition of $BQP$ so all circuits consist only of Clifford gates and at most $k$ $T$-gates).
Is it possible to show that $T_k \subsetneq T_{k+1}$ for each $k$? In particular, can we show that $T_0 \subsetneq T_1$?