Consider a nonnegative operator $T$ (as in all its eigenvalues are nonnegative) on a $d$-dimensional Hilbert space with rank $r < d$ so that its nullity is $n = d - r$. Let us use a standard basis $e_1, \dots, e_d$ so that $T_{ij}$ denotes its matrix representation. Suppose that we crop the matrix $T$ to form a new matrix $T_k$, in the following way, i.e., $T_k$ is a $(n+k)$-square matrix with entries $T_{ij}$ for $1 \le i,j \le n+k$.
Suppose that the projection operator onto $\ker T$ is irreducible as a matrix operator with respect to the standard basis. Is it true that
$$ \ker T_k=n, \quad\forall k \le n $$
Some background. This claim seems to be used in this paper (Mielke, Andreas. Ferromagnetism in the Hubbard model and Hund's rule. Physics Letters A 174, no. 5-6 (1993): 443-448.) (see Eq. 16), but since it's a mathematical physics paper, I attempted to omit all the unnecessary stuff in my question.