The pinnacle-set enumeration conjecture for signed permutations

Let $T(n,k)$ be the sequence defined by

$$T(n,0)=T(n,n)=1$$

and, for $0<k<n$,

$$T(n,k)=2T(n-1,k-1)+T(n-1,k).$$

For each positive integer $n$, let $\mathsf{APS}^B_n$ denote the set of admissible pinnacle sets for signed permutations of size $n$. Pinnacle-set enumeration conjecture.

$$\left|\mathsf{APS}^B_n\right|=T\left(n,\left\lfloor\frac{n-1}{2}\right\rfloor\right).$$

This conjecture proposes a direct correspondence between the number of admissible pinnacle sets in the hyperoctahedral group and the indicated entries of OEIS sequence A119258. The paper presents it as a possible connection and does not establish the identity in general.