Product identities for the four dual Nahm sums of Zagier's Example 12

For $|q|<1$, define $J_m=(q^m;q^m)_\infty$ and $J_{a,m}=(q^a,q^{m-a},q^m;q^m)_\infty$. Let the two dual Nahm sums be the two triple sums displayed below, with denominator factors $(q^5;q^5)_i(q^5;q^5)_j(q^5;q^5)_k$. Example 12 dual identity conjecture. The two displayed Nahm sums equal the stated combinations of infinite-product quotients:

$$\text{the first displayed triple sum}=2\frac{J_{50}^{11}}{J_{5,50}^4J_{10,50}J_{15,50}^3J_{20,50}^2J_{25}}+q\frac{J_{50}^{11}J_{10,50}}{J_{5,50}^5J_{15,50}^2J_{20,50}^4J_{25}}+4q^2\frac{J_{50}^{12}}{J_{5,50}^3J_{10,50}^2J_{15,50}^3J_{20,50}J_{25}^3},$$ $$\text{the second displayed triple sum}=\frac{J_{20,50}J_{50}^{11}}{J_{5,50}^2J_{10,50}^4J_{15,50}^5J_{25}}+2q\frac{J_{50}^{11}}{J_{5,50}^3J_{10,50}^2J_{15,50}^4J_{20,50}J_{25}}+4q^4\frac{J_{50}^{12}}{J_{5,50}^3J_{10,50}J_{15,50}^3J_{20,50}^2J_{25}^3}.$$

These product representations are proposed to justify modularity of the dual of Zagier's Example 12; the source does not report a proof.