Constant lower-bound conjecture for extremal rank-one POVM simulation

Let $\mathbf{M}=(M_1,\ldots,M_n)$ be an arbitrary extremal rank-one POVM on $\mathbb{C}^d$. A partition $\{X_\gamma\}_{\gamma=1}^\alpha$ of $[n]$ satisfies $|X_\gamma|\leq d-1$ when every part has at most $d-1$ elements. Let $q_{\mathrm{succ}}$ be the success probability associated with such a partition in the proposed simulation scheme. Constant lower-bound conjecture. For every extremal rank-one POVM $\mathbf{M}$ on $\mathbb{C}^d$, there exists a partition $\{X_\gamma\}_{\gamma=1}^\alpha$ of $[n]$ satisfying $|X_\gamma|\leq d-1$ such that the corresponding value of $q_{\mathrm{succ}}$ is larger than a positive constant independent of $d$. This is the analytical and numerical formulation used to support a dimension-independent success probability for simulating extremal rank-one measurements; the source gives no resolution.