Kernel equality conjecture for the extremal noncommutative power-sum bound

Let $n,d\ge 2$. For hermitian operators $X_1,\dots,X_n$ on a Hilbert space, define the extremal difference

$$H_{2d}(X_1,\dots,X_n)-\mu_{n,d}(X_1^{2d}+\cdots+X_n^{2d}).$$

Kernel equality conjecture. For every such tuple,

$$\ker \left(H_{2d}(X_1,\dots,X_n)-\mu_{n,d}(X_1^{2d}+\cdots+X_n^{2d})\right) = \ker X_1\cap\cdots\cap \ker X_n.$$

The equality is established in the paper for the quartic two-variable case discussed immediately beforehand, but the general assertion is presented as a speculation. If true, it would imply that equality in the corresponding extremal operator inequality occurs only when all operators vanish, and would yield a strictly improved bound for each fixed finite-dimensional tuple.