Exact tensor rank of the 5 × 5 determinant tensor

Determine the exact tensor rank of the 5 × 5 determinant tensor \(\operatorname{det}_5\), which is currently bounded by \(27 \leq \mathbf{R}(\operatorname{det}_5) \leq 42\).

Background

The paper states that the tensor ranks of the 3 × 3 and 4 × 4 determinant tensors are known exactly, with values 5 and 12, respectively. For the 5 × 5 determinant tensor, the results cited in the paper provide only the lower and upper bounds $27 \leq \mathbf{R}(\operatorname{det}_5) \leq 42$. Determining the precise value is therefore left unresolved.