Conjecture on the minimizer of the auxiliary matrix bound

Let $L(g)$ be the Kronecker product of $g$ copies of the matrix $M^+(1)$, and let $h_0$ be the quantity defined by

$$h_0=\min\{k\geq 2^g+\operatorname{rank}(S): S\text{ is a principal submatrix of }B\text{ of order }k\}.$$

Here $B$ is the matrix appearing in the preceding proposition. Minimizer conjecture. The number $h_0$ is reached at $L(g)$. If true, this would imply the bound $\Theta(2)\leq 4^g-3^g$ for all $g$. The source gives no resolution of this matrix-theoretic conjecture.