Strong induced log-concavity conjecture for equivariant thagomizer Kazhdan–Lusztig polynomials

Fix $n\ge 0$, and let $T_n$ be the thagomizer matroid with the action of the hyperoctahedral group $\mathcal{B}_n$. Write $P_{T_n}^{\mathcal{B}_n}(t)$ and $Q_{T_n}^{\mathcal{B}_n}(t)$ for its equivariant Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials, respectively. For coefficient sequences, use the specialization $W=\mathcal{B}_n$, $G=\mathcal{B}_{2n}$, and $H=\mathcal{B}_n\times\mathcal{B}_n$ in Definition~; equivalently, set

$$\mathsf{P}_{n,k}(X,Y)=[t^k]\mathsf{P}_n(X,Y;t),\qquad \mathsf{Q}_{n,k}(X,Y)=[t^k]\mathsf{Q}_n(X,Y;t),$$

with $\mathsf{P}_{n,k}=\mathsf{Q}_{n,k}=0$ for $k>\lfloor n/2\rfloor$. Strong induced log-concavity conjecture. For every $n\ge 0$, the coefficient sequences of $P_{T_n}^{\mathcal{B}_n}(t)$ and $Q_{T_n}^{\mathcal{B}_n}(t)$ are strongly induced log-concave: for all $1\le i\le j$, the differences

$$\mathsf{P}_{n,i}(X,Y)\mathsf{P}_{n,j}(X,Y)-\mathsf{P}_{n,i-1}(X,Y)\mathsf{P}_{n,j+1}(X,Y)$$

and the analogous differences with $\mathsf{P}$ replaced by $\mathsf{Q}$ are Schur positive in $\Lambda[X,Y]$. This conjectures simultaneous strong induced log-concavity for both equivariant polynomial sequences. The property strengthens ordinary induced log-concavity, which is the case $i=j$, and is not automatic from honesty of the individual coefficient representations. Its resolution would establish a refined positivity phenomenon for equivariant Kazhdan–Lusztig theory of thagomizer matroids with hyperoctahedral symmetry.