Nesting conjecture for primitive normalized Witten intersection numbers

For $g\geq2$, $n\geq1$, and $d_1,\ldots,d_n\geq1$ satisfying $|\mathbf d|=3g-3+n$, let $C(\mathbf d)$ denote the primitive normalized Witten intersection number, and write $C(3g-2)$ and $C(2^{3g-3})$ for the corresponding one-part and repeated-argument values. Nesting conjecture.

$$C(3g-2)\leq C(\mathbf d)\leq C(2^{3g-3}).$$

The claim was motivated by numerical data for fixed genus, but the paper gives a counterexample: $C(4,4)<C(3,5)$ contradicts the proposed ordering. Thus the conjecture is refuted.