Homological distance product formula for arbitrary finite-field chain complexes

Let $F$ be a finite field, and let $\mathcal{A}$ and $\mathcal{B}$ be bounded chain complexes of vector spaces over $F$. Denote by $d_i(\mathcal{A})$, $d_i(\mathcal{B})$, and $d_j(\mathcal{A}\times\mathcal{B})$ their homological distances at the indicated levels. The product-formula identity is

$$d_j(\mathcal{A}\times\mathcal{B})=\min_{i\in\mathbb{Z}}d_i(\mathcal{A})d_{j-i}(\mathcal{B}).$$

Arbitrary finite-field homological distance conjecture. For any pair of bounded chain complexes of vector spaces over a finite field, the homological distances $d_j(\mathcal{A}\times\mathcal{B})$ are given by this identity.

The claim is presented after analytical results for products involving one-dimensional complexes and numerical tests over several finite fields. The supplied status evidence reports a counterexample discovered by Yaroslav Shitov, so the conjecture is refuted.