Combinatorial reciprocity conjecture for the determinants $D_{s,t}(n)$

Let $D_{s,t}(n)$ denote the determinant associated with the corresponding family of cyclically symmetric rhombus-tiling regions, with the parity of $s-t$ distinguishing ordinary tiling counts from weighted counts. Combinatorial reciprocity conjecture. For integers $n\geqslant r\geqslant1$,

$$D_{2r-1,0}(2n+1)=D_{0,0}(2n-2r+2)\big|_{\mu\to1-\mu-6n}.$$

This conjecture proposes a reciprocity between determinants that count cyclically symmetric rhombus tilings and determinants that perform weighted counts. For concrete integer parameters, one determinant may fall outside the stated combinatorial interpretation, for example when the hole is larger than the hexagon.