Binomial generating-series conjecture for orthogonal vector spaces

In the orthogonal vector-space situation, let $\binom{x}{n}$ denote the generalized binomial coefficient, let $\boldsymbol{D}_n$ and $\boldsymbol{B}_n$ denote the corresponding Dynkin types, and let $\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_d$ and $\mathrm{DT}^{\mathrm{sd}}_d$ be the numerical self-dual invariants. Binomial generating-series conjecture. For every integer $n\geqslant 0$,

$$\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{2n}=\binom{1/4}{n},\qquad \mathrm{DT}^{\mathrm{sd}}_{2n}=(-1)^n\binom{1/4}{n},$$ $$\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{2n+1}=\binom{-1/4}{n},\qquad \mathrm{DT}^{\mathrm{sd}}_{2n+1}=(-1)^n\binom{-1/4}{n}.$$

Equivalently,

$$\sum_{n=0}^{\infty}\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{\boldsymbol{D}_n}q^n=(1+q)^{1/4},\qquad \sum_{n=0}^{\infty}\mathrm{DT}^{\mathrm{sd}}_{\boldsymbol{D}_n}q^n=(1-q)^{1/4},$$ $$\sum_{n=0}^{\infty}\frac{\chi}{\mathrm{J}}^{\mathrm{sd}}_{\boldsymbol{B}_n}q^n=(1+q)^{-1/4},\qquad \sum_{n=0}^{\infty}\mathrm{DT}^{\mathrm{sd}}_{\boldsymbol{B}_n}q^n=(1-q)^{-1/4}.$$

These formulas extend the observed numerical pattern for orthogonal vector spaces; their status is not resolved in the supplied source.