Conjectural fourth relation among rank-four tadpole Nahm sums

For $h=1,2,3,4$, let $A^{(h)}(q)$, $B^{(h)}(q)$, $C^{(h)}(q)$, and $D^{(h)}(q)$ denote the four Nahm sums introduced in the source, and let $a$, $b$, and $R_4^{(h)}$ be the corresponding functions defined there. Fourth relation conjecture.

$$b\cdot D^{(h)}(q)-a\cdot A^{(h)}(q)=R_4^{(h)}.$$

This is the one remaining relation in a system of four; the other three relations are proved in the preceding theorem. The notation and explicit right-hand side are defined elsewhere in the paper.