An inverse semigroup is an algebra with two operations: binary $\cdot$ and unary $^{-1}$ such that $\cdot$ is associative and $xx^{-1}x=x, xx^{-1}yy^{-1}=yy^{-1}xx^{-1}$. The Brandt semigroup with 1, $B_2^1$, is the inverse semigroup of $2\times 2$-matrices consisting of 0, I, and the four matrix units $e_{i,j}$, $i,j=1,2$ where $e_{i,j}$ is the matrix with $(i,j)$-entry 1 and other entries 0, $e_{i,j}^{-1}=e_{j,i}$. It is known (Kleiman) that the identities of $B_2^1$ are not finitely based.

**Question.** Is it known that the identities of any finite inverse semigroup containing $B_2^1$ as an inverse subsemigroup are not finitely based?