Conjecture on generators of the ideal $I^{\mu}_N$ in the parabolic Hecke algebra

Let $H^{\mu}(S_n)$ be the parabolic Hecke algebra, let $I^{\mu}_N$ be the ideal under consideration, and define

$$X^{\mu}_N=e_{\mu}T_{\gamma_{\mu}}C^{\dagger}_{w_{N+1}}T_{\gamma_{\mu}}^{-1}e_{\mu},\qquad Y^{\mu}_N=e_{\mu}C^{\dagger}_{\tilde{w}_{N+1}}e_{\mu}.$$

Generator conjectures. The element $X^{\mu}_N$ generates $I^{\mu}_N$; the element $Y^{\mu}_N$ generates $I^{\mu}_N$; and

$$X^{\mu}_N=Y^{\mu}_N.$$

Statement a) was previously conjectured and proved in several special cases, including $N=2$ for any $\mu$ and arbitrary $N$ for specified types of $\mu$. The present work gives supporting evidence for the equality and proves all three statements in some special cases; if the equality holds, the two generator assertions are equivalent.