when is a system of mass-action reactions guaranteed to satisfy detailed balance?

when does a system of mass-action reactions satisfy detailed balance for *any non-zero choice of the rate constants*?


consider a trivial example:


$X + Y \ce{<-->[k1][k2]} XY$


satisfies detailed balance for any $k\_1, k\_2 > 0$. Here is a less trivial example:


$A + X \ce{<-->[k1][k2]} AX$


$B + X \ce{<-->[k3][k4]} BX$


we can ask if this system satisfies detailed balance, ie whether for any cycle the product of the rates going clockwise must equal the product of the rates going counter-clockwise ([Wegscheider condition](https://en.wikipedia.org/wiki/Detailed_balance#:%7E:text=The%20Wegscheider%20conditions%20demonstrate%20that,all%20states%20far%20from%20equilibrium.))?


It appears that any positive non-zero values of $k\_1,k\_2,k\_3,k\_4$ this system satisfies Wegscheider. Is that right and if so is there a simple property of the *structure* of such reactions that we can check to tell if it will satisfy Wegscheider?


