Conjecture on Geck's and Murphy-type cellular bases for type B Hecke algebras

Fix a weighting $(s_1,s_2)\in \mathbb{Z}^2$ and set

$$\theta=(-s_1,\tfrac{1}{2}-s_2)\in \mathbb{Z}[\tfrac{1}{2}]^2.$$

Consider Geck's cellular Kazhdan–Lusztig-type basis of the type $B$ Hecke algebra and the graded cellular Murphy-type basis constructed in Theorem A. Basis comparison conjecture. The cellular Kazhdan–Lusztig-type basis differs from the graded cellular Murphy-type basis only by multiplication by a unitriangular change-of-basis matrix. The conjecture is motivated by the reconciliation of Lusztig's $a$-function ordering with the authors' $\theta$-dominance ordering and by the unitriangularity of the corresponding decomposition numbers; the analogous statement is known in type $A$, but the type $B$ assertion remains open.