Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is *nilpotent* if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned by all possible products $v_1\ldots v_k$ with $v_i\in V$. The conjecture is:

Assume that both $V^k$ and $V^{k+1}$ are nilpotent spaces. Then, $\dim V^k\geqslant \dim V^{k+1}$.

Some experiments with upper-triangular nilpotent spaces make me hope that this conjecture is true in general. I would be very happy if someone was able to help me with either a proof or a counterexample, even for $k=1$.