The normalized BGW-number nesting conjecture

Let $g\ge2$ and let $\mathbf d$ be a partition of $g-1$. Write $C(\mathbf d)$ for the normalized BGW number associated with $\mathbf d$. Nesting conjecture.

$$C(g-1)\le C(\mathbf d)\le C(1^{g-1}).$$

The inequalities assert that the normalized numbers are bounded by the values for the crudest and finest partitions. They were checked numerically through genus $40$; the assertion for all $g$ remains open.