Type C and type D Chan–Robbins–Yuen volume conjecture

Let $CRYC_{n+1}$ and $CRYD_{n+1}$ denote the type $C_{n+1}$ and type $D_{n+1}$ analogues of the Chan–Robbins–Yuen polytope $CRYA_n$. For a nonnegative integer $k$, let

$$Cat(k)=\frac{1}{k+1}\binom{2k}{k}$$

be the $k$th Catalan number. Type C and type D Chan–Robbins–Yuen volume conjecture. The normalized volumes satisfy

$$\operatorname{vol}(CRYC_{n+1})=2^{(n-1)^2+n}\prod_{k=0}^{n-1}Cat(k),$$ $$\operatorname{vol}(CRYD_{n+1})=2^{(n-1)^2}\prod_{k=0}^{n-1}Cat(k).$$

These polytopes are introduced as analogues of $CRYA_n$; the paper proves their vertex counts, but the displayed volume formulas are conjectural and no resolution is given.