Uniform boundedness conjecture for the suboptimality ratio of projective measurements

Let $D(\mathcal{S})$ denote the density operators supported on an $r$-dimensional subspace $\mathcal{S}\subseteq\mathbb{C}^n$, and let $K_{n,r}$ be the suboptimality ratio defined for optimizing the quantity in the paper's maximization problem over projective measurements on $\mathbb{C}^n$ versus those aligned with $\mathcal{S}$. Uniform boundedness conjecture. There exists $C \geq 1$ such that

$$K_{n,r} \leq C$$

for all $n,r \geq 1$. This conjecture asserts that measurements not fully aligned with the low-rank subspace provide at most a universal constant-factor improvement; proving it would justify restricting the optimization to measurements aligned with $\mathcal{S}$, especially when the states have low rank.