The conjectured formula for $d54d_w$ for special permutations

Let $w$ be a special permutation with code

$$\operatorname{code}(w)=(n,*,n-1,*,n-2,\dots,*,2,*,1,0,0,\dots),$$

where each $*$ is either $0$ or empty. Let $a_1<\cdots<a_k=n+k$ be the positions of the zeros in the code that are preceded by a nonzero entry. Define

$$a_\delta(y_1,\dots,y_n)=\prod_{1\leq i<j\leq n}(y_i-y_j).$$

An $n$-element subset $J=\{j_1,\dots,j_n\}$ of $\{1,2,\dots,n+k\}$ is valid if

$$\#(J\cap\{a_{i-1}+1,a_{i-1}+2,\dots,a_i\})=a_i-a_{i-1}-1$$

for $1\leq i\leq k$, with $a_0=0$, and let $\varepsilon_J=(-1)^{d_J}$, where

$$d_J=\binom{n+k+1}{2}-1-(a_1+1)-\cdots-(a_{k-1}+1)-\sum_{i\in J}i.$$

The conjectured formula for $\mathfrak{D}_w$. For special $w$,

$$\mathfrak{D}_w=C_{nk}\sum_{J=\{j_1,\dots,j_k\}}\varepsilon_J\,a_\delta(y_{n+k-j_1+1},y_{n+k-j_2+1},\dots,y_{n+k-j_k+1}),$$

where

$$C_{nk}=\frac{(n+1)!(n+2)!\cdots(n+k-1)!}{\binom{n+1}{2}!}$$

and $J$ ranges over all valid subsets of $\{1,2,\dots,n+k\}$. This gives a proposed explicit value for the polynomial $\mathfrak{D}_w$ associated with a special permutation.