Type B, C and D flow-polytope volume relations

Let $K_n^D$, $K_n^B$ and $K_n^C$ be the signed complete graphs whose edges correspond to the positive roots of types $D_n$, $B_n$ and $C_n$, respectively. Write $\mathcal{F}_{K_n^X}(a_1,\ldots,a_n)$ for the corresponding flow polytope, and let $CRYD_n=\mathcal{F}_{K_n^D}(2,0,\ldots,0)$. Type B, C and D flow-polytope volume conjecture. The following relations hold:

$$\operatorname{vol}\mathcal{F}_{K_n^C}(2,0,\ldots,0)=2^{n-2}\operatorname{vol}(CRYD_n),$$

and, except when $n=2$, where $\operatorname{vol}\mathcal{F}_{K_n^D}(2,0)=\operatorname{vol}\mathcal{F}_{K_n^D}(1,1)$,

$$\operatorname{vol}\mathcal{F}_{K_n^{\{B,C,D\}}}(2,0,\ldots,0)=2\operatorname{vol}\mathcal{F}_{K_n^{\{B,C,D\}}}(1,1,0,\ldots,0).$$

These identities are presented after numerical observations for several families of signed-graph flow polytopes; the source does not provide proofs or a resolution.