The shortest-identity conjecture for the transformation semigroup T_5

Let $T_5$ be the full transformation semigroup on a set of five elements, and let

$$(xy)^{3+\operatorname{lcm}(4)}(yx)^5(xy)^4 \equiv_5 (xy)^3(yx)^5(xy)^{4+\operatorname{lcm}(4)}.$$

Shortest-identity conjecture for $T_5$. This identity is the shortest identity of $T_5$. The preceding theorem establishes that this identity holds, while the conjecture concerns its minimality; the paper notes that exhaustive search verifies the analogous minimality claim only for $k\leq 4$.