Unbounded strength of the determinant

For each form $P$, let $\str(P)$ denote the least number of reducible forms whose sum is $P$. Let $\mathrm{det}_n$ be the determinant viewed as a homogeneous polynomial in the rows (or columns) of an $n\times n$ matrix. Strength conjecture.

$$\lim_{n \to \infty} \str(\mathrm{det}_n) = \infty.$$

The strength measures expansions in which the factors need not be multilinear. Even obtaining any nontrivial bound for the strength of the determinant is open, so this conjecture asks for the much weaker assertion that the strength is unbounded as the matrix size grows.