Strong USP capacity conjecture

A strong USP is a collection of strings with the strong uniquely solvable puzzle property, and the strong USP capacity is the largest constant $C$ such that there exist strong USPs of size $(C-o(1))^k$ and width $k$ for infinitely many values of $k$. The USP capacity equals $3/2^{2/3}$.

Strong USP capacity conjecture. The strong USP capacity equals

$$3/2^{2/3}.$$

This would show that strong USPs attain the same asymptotic capacity as USPs and, via the associated group-theoretic construction, would yield matrix multiplication exponent $\omega=2$. The paper gives a lower bound of $2^{2/3}$ for the strong USP capacity but no matching upper bound beyond the general USP-capacity bound.