Strict exclusion conjecture for adjoining zero to positive sets

Let $k\geq 1$, let $B\subset\mathbb{R}_{>0}$ be a set with $|B|=k-1$, and define $r_k:=\min\{|\Omega_2(A)|:|A|=k\}$, where

$$\Omega_2(A)=\{\rho(M):M\in A^{2\times 2}\}.$$

Strict exclusion conjecture. One has

$$|\Omega_2(B\cup\{0\})|<r_k.$$

This conjecture concerns whether sets containing zero can attain the minimum predicted by the preceding computational conjecture; its status is not resolved in the source.