Conjecture on the minimum size of uniqueness sets for the Ising model

Let $u(k,q)$ denote the size of the smallest set of uniqueness for the positive function class $\left(\mathcal{B}^{k}_{q}\right)_+$. In particular, consider the case $q=2$. Minimum-size uniqueness conjecture.

$$u(k,2)=k+1,$$

i.e., for every $k$ there are no sets of uniqueness having at most $k$ elements. This conjecture concerns the exact minimum cardinality of a set of uniqueness in the $q=2$ case; the preceding results establish corresponding bounds, while the equality asserted here remains unresolved in the supplied text.